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Karl  Friedrich  Gauss 


General  Investigations 


Curved  Surfaces 


1827  AND  1825 


TRANSLATED    WITH    NOTES 

AND   A 

BIBLIOGRAPHY 

BY 

JAMES  CADDALL  MOREHEAD,  A.M.,  M.S.,  and  ADAM   MILLER   HILTEBEITEL,    A.M. 

J.    S.   K.  FELLOWS    IN    MATHEMATICS   IN    PRINCETON    UNIVERSITY 


THE   PRINCETON    UNIVERSITY   LIBRARY 
1902 


Copyright,  1902,  by 
The  Princeton  University  Library 


BOSTON  COLLEGE  LVBR^^ 
^ChStNV3T  hill.  MASS. 

241441 


C  S.  Robinson  &  Co.,   University  Press 
Princeton,  N.  J. 


INTRODUCTION 

In  1827  Gauss  presented  to  the  Royal  Society  of  Gottingen  his  important  paper  on 
the  theory  of  surfaces,  which  seventy-three  years  afterward  the  eminent  French 
geometer,  who  has  done  more  than  any  one  else  to  propagate  these  principles,  charac- 
terizes as  one  of  Gauss's  chief  titles  to  fame,  and  as  still  the  most  finished  and  use- 
ful introduction  to  the  study  of  infinitesimal  geometry \  This  memoir  may  be  called: 
General  Investigations  of  Curved  Surfaces,  or  the  Paper  of  1827,  to  distinguish  it 
from  the  original  draft  written  out  in  1825,  but  not  published  until  1900.  A  list  of 
the  editions  and  translations  of  the  Paper  of  1827  follows.  There  are  three  editions 
in  Latin,  two  translations  into  French,  and  two  into  German.  The  paper  was  origin- 
ally published  in  Latin  under  the  title  : 

I  a.     Disquisitiones  generales  circa  superficies  curvas 
auctore  Carolo  Friderico  Gauss 

Societati  regise  oblatse  D.  8.  Octob.  1827, 

and  was  printed  in :  Commentationes  societatis  regise  scientiarum  Gottingensis  recen- 
tiores,  Commentationes  classis  mathematicse.  Tom.  VI.  (ad  a.  1823-1827).  Gottingse, 
1828,  pages  99-146.  This  sixth  volume  is  rare ;  so  much  so,  indeed,  that  the  British 
Museum  Catalogue  indicates  that  it  is  missing  in  that  collection.  With  the  signatures 
changed,  and  the  paging  changed  to  pages  1-50,  la  also  appears  with  the  title  page 
added  : 

I  b.     Disquisitiones  generales  circa  superficies  curvas 
auctore  Carolo  Friderico  Gauss. 

Gottingse.     Typis  Dieterichianis.     1828. 

II.  In  Monge's  Application  de  I'analyse  a  la  geometric,  fifth  edition,  edited  by 
Liouville,  Paris,  1850,  on  pages  505-546,  is  a  reprint,  added  by  the  Editor,  in  Latin 
under  the  title:  Recherches  sur  la  theorie  generale  des  surfaces  courbes;  Par  M. 
C.-F.  Gauss. 


•  G.  Darboux,  Bulletin  des  Sciences  Math.    Ser.  2,  vol.  24,  page  278,  1900, 


iv  INTRODIJCTION" 

Ilia.  A  third  Latin  edition  of  this  paper  stands  in:  Gauss,  Werke,  Herausge- 
geben  von  der  Koniglichen  Gesellschaft  der  Wissenschaften  zu  Gottingen,  Vol.  4,  Got- 
tingen,  1873,  pages  217-258,  witliout  change  of  the  title  of  the  original  paper  (la). 

Ill  5.  The  same,  without  change,  in  Vol.  4  of  Gauss,  Werke,  Zweiter  Abdruck, 
Gottingen,  1880. 

IV.  A  French  translation  was  made  from  Liouville's  edition,  II,  by  Captain 
Tiburce  Abadie,  ancien  eleve  de  I'Ecole  Poly  technique,  and  appears  in  Nouvelles 
Annales  de  Mathematique,  Vol.  11,  Paris,  1852,  pages  195-252,  under  the  title : 
Recherches  generales  sur  les  surfaces  courbes ;  Par  M.  Gauss.  This  latter  also 
appears  under  its  own  title. 

Vfl.  Another  French  translation  is  :  Recherches  Generales  sur  les  Surfaces 
Courbes.  Par  M.  C.-F.  Gauss,  traduites  en  fran<5'ais,  suivies  de  notes  et  d'<^tudes 
sur  divers  points  de  la  Theorie  des  Surfaces  et  sur  certaines  classes  de  Courbes,  par 
M.  E.  Roger,  Paris,  1855. 

Yb.  The  same.  Deuxieme  Edition.  Grenoble  (or  Paris),  1870  (or  1871),  160 
pages. 

VI.  A  German  translation  is  the  first  portion  of  the  second  part,  namely,  pages 
198-232,  of:  Otto  Boklen,  Analytische  Geometrie  des  Raumes,  Zweite  Auflage,  Stutt- 
gart, 1884,  under  the  title  (on  page  198)  :  Untersuchungen  liber  die  allgemeine  Theorie 
der  krummen  Flachen.  Von  C.  F.  Gauss.  On  the  title  page  of  the  book  the  second 
part  stands  as  :  Disquisitiones  generales  circa  superficies  curvas  von  C.  F.  Gauss,  ins 
Deutsche  iibertragen  mit  Anwendungen  und  Zusiitzen  .... 

VII «.  A  second  German  translation  is  No.  5  of  Ostwald's  Klassiker  der  exacten 
Wissenschaften :  Allgemeine  Flachentheorie  (Disquisitiones  generales  circa  superficies 
curvas)  von  Carl  Friedrich  Gauss,  (1827).  Deutsch  herausgegeben  von  A.  Wangerin. 
Leipzig,  1889.     62  pages. 

VII 3.     The  same.      Zweite  revidirte  Auflage.     Leipzig,  1900.      64  pages. 

The  English  translation  of  the  Paper  of  1827  here  given  is  from  a  copy  of  the 
original  paper,  I  a ;  but  in  the  preparation  of  the  translation  and  the  notes  all  the 
other  editions,  except  V«,  were  at  hand,  and  were  used.  The  excellent  edition  of 
Professor  Wangerin,  VII,  has  been  used  throughout  most  freely  for  the  text  and 
notes,  even  when  special  notice  of  this  is  not  made.  It  has  been  the  endeavor  of 
the  translators  to  retain  as  far  as  possible  the  notation,  the  form  and  punctuation  of 
the  formulae,  and  the  general  style  of  the  original  papers.  Some  changes  have  been 
made  in  order  to  conform  to  more  recent  notations,  and  the  most  important  of  those 
are  mentioned  in  the  notes. 


INTKODUCTIOl^  T 

The  second  paper,  the  translation  of  which  is  here  given,  is  the  abstract  (Anzeige) 
which  Gauss  presented  in  German  to  the  Royal  Society  of  Gottingen,  and  which  was 
published  in  the  Gottingische  gelehrte  Anzeigen.  Stuck  177.  Pages  1761-1768.  1827. 
November  5.  It  has  been  translated  into  English  from  pages  341-347  of  the  fourth 
volume  of  Gauss's  Works.  This  abstract  is  in  the  nature  of  a  note  on  the  Paper  of 
1827,  and  is  printed  before  the  notes  on  that  paper. 

Recently  the  eighth  volume  of  Gauss's  Works  has  appeared.  This  contains  on 
pages  408-442  the  paper  which  Gauss  wrote  out,  but  did  not  publish,  in  1825.  This 
paper  may  be  called  the  New  General  Investigations  of  Curved  Surfaces,  or  the  Paper 
of  1825,  to  distinguish  it  from  the  Paper  of  1827.  The  Paper  of  1825  shows  the 
manner  in  which  many  of  the  ideas  were  evolved,  and  while  incomplete  and  in  some 
cases  inconsistent,  nevertheless,  when  taken  in  connection  with  the  Paper  of  1827, 
shows  the  development  of  these  ideas  in  the  mind  of  Gauss.  In  both  papers  are 
found  the  method  of  the  spherical  representation,  and,  as  types,  the  three  important 
theorems  :  The  measure  of  curvature  is  equal  to  the  product  of  the  reciprocals  of  the 
principal  radii  of  curvature  of  the  surface,  The  measure  of  curvature  remains  unchanged 
by  a  mere  bending  of  the  surface,  The  excess  of  the  sum  of  the  angles  of  a  geodesic 
triangle  is  measured  by  the  area  of  the  corresponding  triangle  on  the  auxiliary  sphere. 
But  in  the  Paper  of  1825  the  first  six  sections,  more  than  one-fifth  of  the  whole  paper, 
take  up  the  consideration  of  theorems  on  curvature  in  a  plane,  as  an  introduction, 
before  the  ideas  are  used  in  space  ;  whereas  the  Paper  of  1827  takes  up  these  ideas 
for  space  only.  Moreover,  while  Gauss  introduces  the  geodesic  polar  coordinates  in 
the  Paper  of  1825,  in  the  Paper  of  1827  he  uses  the  general  coordinates,  ^?,  q,  thus 
introducing  a  new  method,  as  well  as  employing  the  principles  used  by  Monge  and 
others. 

The  publication  of  this  translation  has  been  made  possible  by  the  liberality  of 
the  Princeton  Library  Publishing  Association  and  of  the  Alumni  of  the  University 
who  founded  the  Mathematical  Seminary. 

H.  D.  Thompson. 


Mathematical  Seminary, 

Princeton  University  Library, 

January  29,  1902. 


CONTENTS 

PAGE 

Gauss's  Paper  of  1827,  General  Investigations  of  Curved  Surfaces    ...  1 

Gauss's  Abstract  of  the  Paper  of  1827     ........  45 

Notes  on  the  Paper  of  1827 51 

Gauss's  Paper  of  1825,  New  General  Investigations  of  Curved  Surfaces          .  79 

Notes  on  the  Paper  of  1825 Ill 

Bibliography  of  the  General  Theory  of  Surfaces        ......  115 


DISQUISITIONES    GENERALES 


CIRCA 


SUPERFICIES    CURVAS 


AUCTORE 


CAROLO    FRIDERICO    GAUSS 


SOCIETATI    REGIAE    OBLATAE    D.    8.    OCTOB.    1827 


COMMENTATIONES    SOCIETATIS    REGIAE    SCIENTIARUM 
GOTTINGENSIS  RECENTIORES.     VOL.  VI.     GOTTINGAE   MDCCCXXVIII 


GOTTINGAE 

TYPIS    DIETERICHIANIS 
MDCCCXXVIII 


GENERAL   INVESTIGATIONS 

OF 

CURVED    SURFACES 

BY 

KARL  FRIEDRICH   GAUSS 

PRESENTED  TO  THE  ROYAL   SOCIETY,  OCTOBER  8,  1827 


Investigations,  in  which  the  directions  of  various  straight  lines  in  space  are  to  be  /j   p 


'a^f* 


.■i-oJ\ 


considered,  attain  a  high  degree  of  clearness  and  simplicity  if  we  employ,  as  an  auxil-  ^'  •'j'  .       ' 

iary,  a  sphere   of  unit   radius  described   about   an   arbitrary  centre,  and   suppose   the  ^a^-  "-""^^ 

different  points  of  the  sphere  to  represent  the  directions  of  straight  lines  parallel  to  I p,  ^  ^o 

the  radii  ending  at  these  points.  As  the  position  of  every  point  ia  space  is  deter- 
mined by  three  coordinates,  that  is  to  say,  the  distances  of  the  point  from  three  mutually 
perpendicular  fixed  planes,  it  is  necessary  to  consider,  first  of  all,  the  directions  of  the 
axes  perpendicular  to  these  planes.  The  points  on  the  sphere,  which  represent  these 
directions,  we  shall  denote  by  (1),  (2),  (3).  The  distance  of  any  one  of  these  points 
from  either  of  the  other  two  wiU  be  a  quadrant;  and  we  shall  suppose  that  the  direc- 
tions of  the  axes  are  those  in  which  the  corresponding  coordinates  increase. 


It  wiU  be  advantageous  to  bring  together  here  some  propositions  which  are  fre- 
quently used  in  questions  of  this  kind. 

I.  The   angle  between   two    intersecting   straight   lines   is   measured   by   the   arc 
between  the  points  on  the  sphere  which  correspond  to  the  dii-ections  of  the  lines. 

II.  The  orientation  of  any  plane  whatever  can  be  represented  by  the  great  circle 
on  the  sphere,  the  plane  of  which  is  parallel  to  the  given  plane. 


4  KARL  FRIEDRICH  GAUSS 

III.  The  angle  between  two  planes  is  equal  to  the  spherical  angle  between  the 
great  circles  representing  them,  and,  consequently,  is  also  measured  by  the  arc  inter- 
cepted between  the  poles  of  these  great  circles.  And,  in  like  manner,  the  angle  of  inclina- 
tion of  a  straight  line  to  a  plane  is  measured  by  the  arc  drawn  from  the  point  which 
corresponds  to  the  direction  of  the  line,  perpendicular  to  the  great  circle  which  repre- 
sents the  orientation  of  the  plane. 

IV.  Letting  x,  y,  s ;  x',  y' ,  s'  denote  the  coordinates  of  two  points,  r  the  distance 
between  them,  and  L  the  point  on  the  sphere  which  represents  the  direction  of  the  line 
drawn  from  the  first  point  to  the  second,  we  shall  have 

x'=^  X  +  r  cos,  {1)L 
y'—y  +  rcos  (2)Z 
s'=  s  +  r  cos  (3)Z  » 

V.  From  this  it  follows  at  once  that,  generally, 

cos''  (l)i  +  cos'  {2)L  +  cos'  (3)Z  =  1 
and  also,  if  L'  denote  any  other  point  on  the  sphere, 

cos  (1)Z  .  cos  (1)X'+  cos  {2)L  .  cos  (2)X'+  cos  (3)Z  .  cos  (3)X'=  cos  LL'. 

VI.  Theorem.  If  L,  L',  L",  L'"  denote  four  points  on  the  sphere,  and  A  the  angle 
which  the  arcs  LL',  L"L"'  make  at  their  point  of  intersection,  then  we  shall  have 

cos  LL" .  cos  L'L'"—  cos  LL'" .  cos  L'L"^  sin  LL' .  sin  L"L'" .  cos  A 

Demonstration.     Let  A  denote  also  the  point  of  intersection  itself,  and  set 
AL  =  t,    AL'=t',    AL"=^t",    AL'"=t'" 


Then  we  shaU  have 


-jv^ 


^-^^.1 


cos  LL"  =  cos  ^ .  cos  t"  +  sin  t  sin  t"  cos  A 
cos  L'L'"  =  cos  f  cos  t'"  +  sin  t'  sin  f"  cos  A  ^,. j,,  ■  'tv^. 

cos  LL'"  =  cos  t  cos  t"'+  sin  t  sin  t'"  cos  A 
cos  L'L"  —  cos  t'  cos  t"  +  sin  t'  sin  t"  cos  A 
and  consequently, 

cos  LL"  .  cos  L'L'"—  cos  LL'".   cos  L'L" 

=  cos  A  (cos  t  cos  t"  sin  t'  sin  f"  +  cos  f  cos  f"  sin  t  sin  t" 

—  cos  t  cos  t'"  sin  f  sin  t" —  cos  f  cos  t"  sin  t  sin  t'") 
=  cos  A  (cos  t  sin  t' — sin  t  cos  f)  (cos  t"  sin  t'" —  sin  t"  cos  t'") 
=  cos  A  .  sin  {t'—t)  .  sin  {t"'—i") 
=  cosA  .  sin  LL'  .sin  L"L'" 


GEKERAL  ESTVESTIGATIOITS  OF  CURVED  STJEFACES  5 

But  as  there  are  for  each  great  circle  two  branches  going  out  from  the  point  A, 
these  two  branches  form  at  this  point  two  angles  whose  sum  is  180°.  But  our  analysis 
shows  that  those  branches  are  to  be  taken  whose  directions  are  in  the  sense  from  the 
point  Z  to  Z',  and  from  the  point  Z"  to  Z'";  and  since  great  circles  intersect  in  two 
points,  it  is  clear  that  either  of  the  two  points  can  be  chosen  arbitrarily.  Also,  instead 
of  the  angle  A,  we  can  take  the  arc  between  the  poles  of  the  great  circles  of  which  the 
arcs  Z  Z',  Z"  Z'"  are  parts.  But  it  is  evident  that  those  poles  are  to  be  chosen  which 
are  similarly  placed  with  respect  to  these  arcs;  that  is  to  say,  when  we  go  from  Z  to  Z' 
and  from  Z"  to  Z'" ,  both  of  the  two  poles  are  to  be  on  the  right,  or  both  on  the  left. 

VII.     Let  Z,  Z',  Z"  be  the  three  points  on  the  sphere  and  set,  for  brevity, 

cos  {1)Z  =  X,    cos  (2)Z  =  y,    cos  (3)Z  =  z 
cos  (l)X'  =  z',  cos  {2)Z'  =  y',  cos  (3)Z'  =  z' 
cos  (l)i/"=  a;",  cos  {^)Z"=y",  cos  (3)X"=  z" 
and  also 

X  y'  s"  +  x'  y"  z  -\-  x"  y  z'  —  x  y"  z'  —  x'  y  z"  —  x"  y'  z  =i  ^ 

Let  \  denote  the  pole  of  the  great  circle  of  which  ZZ'  is  a  part,  this  pole  being  the  one 
that  is  placed  in  the  same  position  with  respect  to  this  arc  as  the  point  (1)  is  with 
respect  to  the  arc  (2)  (3).     Then  we  shall  have,  by  the  preceding  theorem, 

y  z'  —y'  z  =  cos  (1)\  .  sin  (2)(3)  .  sin  ZZ' , 

or,  because  (2)(3)  =  90°, 

y  z'  — y'  z  =  cos  (l)X.  .  sin  ZZ', 
and  similarly, 

z  x' — z'  X  —  COS  (2)X  .  sin  ZZ' 
xy'  —  x' y  =  COS  (3)X  .  sin  ZZ' 

Multiplying  these  equations  by  x",  y",  z"  respectively,  and  adding,  we  obtain,  by  means 
of  the  second  of  the  theorems  deduced  in  V, 

A  =  cos  X  Z" .  sin  ZZ' 

Now  there  are  three  cases  to  be  distinguished.  First,  when  Z"  lies  on  the  great  cii'cle 
of  which  the  arc  ZZ'  is  a  part,  we  shall  have  Xl*"— 90°,  and  consequently,  A  =  0. 
If  Z"  does  not  lie  on  that  great  circle,  the  second  case  will  be  when  Z"  is  on  the  same 
side  as  X ;  the  third  case  when  they  are  on  opposite  sides.  In  the  last  two  cases  the 
points  Z,  Z',  Z"  will  form  a  spherical  triangle,  and  in  the  second  case  these  points  will  lie 
in  the  same  order  as  the  points  (1),  (2),  (3),  and  in  the  opposite  order  in  the  third  case. 


6  KARL  FRIEDRICH  GAUSS 

Denoting  the  angles  of  this  triangle  simply  by  L,  L',  L"  and  the  perpendicular  drawn  on 
the  sphere  from  the  point  L"  to  the  side  LL'  by  jj,  we  shall  have 

sin^  =  sin  X .  sin  LL" ^  sin  L' .  sin  L'  L", 
and 

\Z"=90°  +p, 

the  upper  sign  being  taken  for  the  second  case,  the  lower  for  the  third.  From  this 
it  follows  that 

i  A  =  sini.  sinXX'.  sinXZ"  =  sini'.  sin  ZX'.  sini'i/"  .    »             \^       n^- ^ 

=  sin  L" .  sin  L L" .  sin  L' L"                                        \''  ;;       '^- Ci^'''  ZJ^ 

Moreover,  it  is  evident  that  the  first  case  can  be  regarded  as  contained  in  the  second  or 
third,  and  it  is  easily  seen  that  the  expression  ±  A  represents  six  times  the  volume  of 
the  pyramid  formed  by  the  points  L,  L',  L"  and  the  centre  of  the  sphere.  Whence, 
finally,  it  is  clear  that  the  expression  ±  ^  A  expresses  generally  the  volume  of  any 
pyramid  contained  between  the  origin  of  coordinates  and  the  three  points  whose  coor- 
■vj  dinates  are  ^,  y,  s  ;  x',  y' ,  z' ;  x" ,  y",  s" . 

3. 

A  curved  surface  is  said  to  possess  continuous  curvature  at  one  of  its  points  A,  if  the 
directions  of  all  the  straight  lines  drawn  from  A  to  points  of  the  surface  at  an  infinitely 
small  distance  from  A  are  deflected  infinitely  little  from  one  and  the  same  plane  passing 
through  A.  This  plane  is  said  to  iouch  the  surface  at  the  point  A.  If  this  condition  is 
not  satisfied  for  any  point,  the  continuity  of  the  curvature  is  here  interrupted,  as  happens, 
for  example,  at  the  vertex  of  a  cone.  The  following  investigations  will  be  restricted  to 
such  surfaces,  or  to  such  parts  of  surfaces,  as  have  the  continuity  of  their  curvature 
nowhere  interrupted.  We  shall  only  observe  now  that  the  methods  used  to  determine 
the  position  of  the  tangent  plane  lose  their  meaning  at  singular  points,  in  which  the 
continuity  of  the  curvature  is  interrupted,  and  must  lead  to  indeterminate  solutions. 

4. 

The  orientation  of  the  tangent  plane  is  most  conveniently  studied  by  means  of  the 
direction  of  the  straight  line  normal  to  the  plane  at  the  point  A,  which  is  also  called  the 
normal  to  the  curved  surface  at  the  point  A.  We  shall  represent  the  direction  of  this 
normal  by  the  point  L  on  the  auxiliary  sphere,  and  we  shall  set 

cos  (1)Z  =X,      cos  (2)Z  =  Y,      cos  (3)X  =Z; 

and  denote  the  coordinates  of  the  point  A  by  x,  y,  z.     Also  let  x  +  dx,  y  +  dy,  z  +  dz 
be  the  coordinates   of  another  point  A'  on  the  curved  surface ;  ds  its  distance  from  A, 


GENEEAL  INVESTIGATIONS  OF  CUEVED  SUEFACES  7 

which  is  infinitely  small ;  and  finally,  let  X  be  the  point  on  the  sphere  representing  the 
direction  of  the  element  ilJ.'.     Then  we  shall  have 

dx  =  ds.  cos  {1)\,       di/  —  ds.Gos{2)\,       ds  =  ds.  cos  {B)\ 

and,  since  X  Z  must  be  equal  to  90°, 

X  cos  (1)X  +  Zcos  (2)X  +  Z  cos  (3)X  =  0 

By  combining  these  equations  we  obtain 

Xdx+Ydi/+Zds  =  0. 

There  are  two  general  methods  for  defining  the  nature  of  a  curved  surface.  The 
Jirst  uses  the  equation  between  the  coordinates  z,  y,  z,  which  we  may  suppose  reduced  to 
the  form  W  =  0,  where  W  will  be  a  function  of  the  indeterminates  x,  ij,  z.  Let  the  com- 
plete differential  of  the  function  W  be 

dW^P  dx-^  Qd>/  +  Rdz 

and  on  the  curved  surface  we  shall  have 

P  dx+  Qdy^  Rds^^ 
and  consequently, 

P  cos  (1)X  +  Q  cos  (2)X  +  R  cos  (3)X  =  0 

Since  this  equation,  as  well  as  the  one  we  have  established  above,  must  be  true  for  the 
directions  of  all  elements  ds  on  the  curved  surface,  we  easily  see  that  X,  Y,  Z  must  be 
proportional  to  P,  Q,  R  respectively,  and  consequently,  since 

X^+Y^+  Z^=^l, 
we  shall  have  either 

_  P Q R 

X  _  /    /  Tii    I      7^2      i       fw\'  Y  ,//D2_l_    /n2_L     r>2\'  Z 


V  {P'+  Q'  +  R')  V{P'+  Q'+  R')  V{P'+  Q'  +  R') 


^  ~  t/  (P^  +  ^^  +  R")        ^       i/  (P^  +  ^^  +  R")        ^      V  {P^+  Q^^-  R') 
The  second  method  expresses  the  coordinates  in  the  form  of  functions  of  two  varia- 
bles, jt?,  q.     Suppose  that  differentiation  of  these  functions  gives 

dz  =z  a  dp  +  a'  dq 
dy  =^h  dp  -\-  b'  dq 
dz  =^c  dp  -\-  c'  dq 


8  KAEL  FREEDEICH  GAUSS 

Substituting  these  values  in  the  formula  given  above,  we  obtain 

{a  X  +  b  Y  +  c  Z)  dp  +  {a'  X  +  b'  Y  +  c'  Z)  dq  =^  0 

Since  this  equation  must  hold  independently   of  the  values  of  the  differentials  djo,  dq, 
we  evidently  shall  have 

aX+b  Y+  c  Z=^,     a'X+b'  Y+c'  Z=0 
From  this  we  see  that  X,  Y,  Z  will  be  proportioned  to  the  quantities 

be'  —  cb',    ca'  —  ac',    aV  —  ba' 
Hence,  on  setting,  for  brevity,  i-     v'j^^ 

V  (^{bc'  —  cb'f  +  {ca'  —  ac'Y  +  {aV  —  ba'Y)  =  A  --• ''       ''J^^^  u^^^ 
we  shall  have  either                              :  \  f^^P 
be'  —  eb'           ea'  —  ae'           ab'  —  ba' 

X=  7  3  Y 7  3  Z T 

AAA 
or 

__  cJ'  —  be'        j^_(f(^' — (^C''        rr_^^'  —  ^^' 
^~         A       '       ^~        A       '      ^~         A 

With  these  two  general  methods  is  associated  a  third,  in  which  one  of  the  coordinates, 
2,  say,  is  expressed  in  the  form  of  a  function  of  the  other  two,  x,  y.  This  method  is 
evidently  only  a  particular  case  either  of  the  first  method,  or  of  the  second.     If  we  set 

dz  zz^tdx  -\-  udy 
we  shall  have  either 

X  =  —TT\     i     rt     i     ~2\ '      Y  =^  _  / 1  -\     i     75    i     Tax '     Z  = 


1/  (1  +  f  +  %"■)      "■         V{\^f-\-v?)  1/  (1  +  i!'  +  M^) 


V/  (1  +  f  +  «')  V  {l^f^U^)  l/  (1  +  ^^  +  M^) 


The  two  solutions  found  in  the  preceding  article  evidently  refer  to  opposite  points  of 
the  sphere,  or  to  opposite  du-ections,  as  one  would  expect,  since  the  normal  may  be  drawn 
toward  either  of  the  two  sides  of  the  curved  surface.  If  we  wish  to  distinguish  between 
the  two  regions  bordering  upon  the  surface,  and  call  one  the  exterior  region  and  the  other 
the  interior  region,  we  can  then  assign  to  each  of  the  two  normals  its  appropriate  solution 
by  aid  of  the  theorem  dcri\'ed  in  Art.  2  (VII),  and  at  the  same  time  establish  a  criterion 
for  ilistinguishing  the  one  region  from  the  other. 


GENERAL  rNTESTIGATIOITS  OF  CUEVED  SURFACES  9 

In  the  first  method,  such  a  criterion  is  to  be  drawn  from  the  sign  of  the  quantity  W. 
Indeed,  generally  speaking,  the  curved  surface  divides  those  regions  of  space  in  which  W 
keeps  a  positive  value  from  those  in  which  the  value  of  W  becomes  negative.  In  fact,  it 
is  easily  seen  from  this  theorem  that,  if  W  takes  a  positive  value  toward  the  exterior 
region,  and  if  the  normal  is  supposed  to  be  drawn  outwardly,  the  first  solution  is  to  be 
taken.  Moreover,  it  will  be  easy  to  decide  in  any  case  whether  the  same  rule  for  the 
sign  of  W  is  to  hold  throughout  the  entire  surface,  or  whether  for  different  parts  there 
will  be  different  rules.  As  long  as  the  coefiicients  P,  Q,  R  have  finite  values  and  do  not 
all  vanish  at  the  same  time,  the  law  of  continuity  will  prevent  any  change. 

If  we  follow  the  second  method,  we  can  imagine  two  systems  of  curved  lines  on  the 
curved  surface,  one  system  for  which  p  is  variable,  q  constant ;  the  other  for  which  q  is 
variable,  jo  constant.  The  respective  positions  of  these  lines  with  reference  to  the  exte- 
rior region  will  decide  which  of  the  two  solutions  must  be  taken.  In  fact,  whenever 
the  three  Unes,  namely,  the  branch  of  the  line  of  the  former  system  going  out  from  the 
point  ^  as  JO  increases,  the  branch  of  the  line  of  the  latter  system  going  out  from  the  point 
.A  as  ^  increases,  and  the  normal  drawn  toward  the  exterior  region,  are  similarly  placed  as 
the  X,  y,  z  axes  respectively  from  the  origin  of  abscissas  (e.  ^.,  if,  both  for  the  former 
three  lines  and  for  the  latter  three,  we  can  conceive  the  first  directed  to  the  left,  the 
second  to  the  right,  and  the  third  upward),  the  first  solution  is  to  be  taken.  But  when- 
ever the  relative  position  of  the  three  lines  is  opposite  to  the  relative  position  of  the 
X,  y,  z  axes,  the  second  solution  will  hold. 

In  the  third  method,  it  is  to  be  seen  whether,  Avhen  z  receives  a  positive  increment,  x 
and  y  remaining  constant,  the  point  crosses  toward  the  exterior  or  the  interior  region. 
In  the  former  case,  for  the  normal  drawn  outward,  the  first  solution  holds ;  in  the  latter 
case,  the  second. 

6. 

Just  as  each  definite  point  on  the  curved  surface  is  made  to  correspond  to  a  definite 
point  on  the  sphere,  by  the  direction  of  the  normal  to  the  curved  surface  which  is  trans- 
ferred to  the  surface  of  the  sphere,  so  also  any  line  whatever,  or  any  figure  whatever,  on 
the  latter  will  be  represented  by  a  corresponding  line  or  figure  on  the  former.  In  the 
comparison  of  two  figures  corresponding  to  one  another  in  this  way,  one  of  which  will  be 
as  thQ  map  of  the  other,  two  important  points  are  to  be  considered,  one  when  quantity 
alone  is  considered,  the  other  when,  disregarding  quantitative  relations,  position  alone 
is  considered. 

The  first  of  these  important  points  will  be  the  basis  of  some  ideas  which  it  seems 
judicious  to  introduce  into  the  theory  of  curved  surfaces.     Thus,  to  each  part  of  a  curved 


1 


.1% 


JX 


10  KARL  FRIEDEICH  GAUSS 

surface  inclosed  within  definite  limits  we  assign  a  total  or  integral  curvature,  which  is 
represented  by  the  area  of  the  figure  on  the  sphere  corresponding  to  it.  From  this 
integral  curvature  must  be  distinguished  the  somewhat  more  specific  curvature  which  we 
shall  caU  the  measure  of  curvature.  The  latter  refers  to  a  point  of  the  surface,  and  shall 
denote  the  quotient  obtained  when  the  integral  curvature  of  the  surface  element  about 
a  point  is  divided  by  the  area  of  the  element  itself;  and  hence  it  denotes  the  ratio  of  the 
infinitely  small  areas  which  correspond  to  one  another  on  the  curved  surface  and  on  the 
sphere.  The  use  of  these  innovations  will  be  abundantly  justified,  as  we  hope,  by  what 
we  shall  explain  below.  As  for  the  terminology,  we  have  thought  it  especially  desirable 
that  aU  ambiguity  be  avoided.  For  this  reason  we  have  not  thought  it  advantageous  to 
follow  strictly  the  analogy  of  the  terminology  commonly  adopted  (though  not  approved  by 
all)  in  the  theory  of  plane  curves,  according  to  which  the  measure  of  curvature  should  be 
called  simply  curvature,  but  the  total  curvature,  the  amplitude.  But  why  not  be  free  in 
the  choice  of  words,  provided  they  are  not  meaningless  and  not  liable  to  a  misleading 
interpretation  ? 

f-i/"^The  position  of  a  figure  on  the  sphere  can  be  either  similar  to  the  position  of  the 
corresponding  figure  on  the  curved  surface,  or  opposite  (inverse).  The  former  is  the  case 
when  two  lines  going  out  on  the  curved  surface  from  the  same  point  in  different,  but  not 
opposite  directions,  are  represented  on  the  sphere  by  lines  similarly  placed,  that  is,  when 
the  map  of  the  line  to  the  right  is  also  to  the  right ;  the  latter  is  the  case  when  the  con- 
trary holds.  We  shall  distinguish  these  two  cases  by  the  positive  or  negative  sign  of  the 
measure  of  curvature.  But  evidently  this  distinction  can  hold  only  when  on  each  surface 
we  choose  a  definite  face  on  which  we  suppose  the  figure  to  lie.  On  the  auxiliary  sphere 
we  shaU  use  always  the  exterior  face,  that  is,  that  turned  away  from  the  centre ;  on  the 
curved  surface  also  there  may  be  taken  for  the  exterior  face  the  one  already  considered, 
or  rather  that  face  from  which  the  normal  is  supposed  to  be  drawn.  For,  evidently,  there 
is  no  change  in  regard  to  the  similitude  of  the  figures,  if  on  the  curved  surface  both  the 
figure  and  the  normal  be  transferred  to  the  opposite  side,  so  long  as  the  image  itself 
is  represented  on  the  same  side  of  the  sphere. 

The  positive  or  negative  sign,  which  we  assign  to  the  measure  of  curvature  accord- 
ing to  the  position  of  the  infinitely  small  figure,  we  extend  also  to  the  integral  curvature 
of  a  finite  figure  on  the  curved  surface.  However,  if  we  wish  to  discuss  the  general  case, 
some  explanations  will  be  necessary,  which  we  can  only  touch  here  briefly.  So  long 
as  the  figure  on  the  curved  surface  is  such  that  to  distinct  points  on  itself  there  corres- 
pond distinct  points  on  the  sphere,  the  definition  needs  no  further  explanation.  But 
whenever  this  condition  is  not  satisfied,  it  will  be  necessary  to  take  into  account  twice 
or  several  times  certain  parts  of  the  figure  on  the  sphere.      Whence  for  a  similar,  or 


GENEEAL  INVESTIGATIONS  OF  CURVED  SURFACES  11 

inverse  position,  may  arise  an  accumulation  of  areas,  or  the  areas  may  partially  or 
wholly  destroy  each  other.  In  such  a  case,  the  simplest  way  is  to  suppose  the  curved 
surface  divided  into  parts,  such  that  each  part,  considered  separately,  satisfies  the  above 
condition ;  to  assign  to  each  of  the  parts  its  integral  curvature,  determining  this  magni- 
tude by  the  area  of  the  corresponding  figure  on  the  sphere,  and  the  sign  by  the  posi- 
tion of  this  figure ;  and,  finally,  to  assign  to  the  total  figure  the  integral  curvature 
arising  from  the  addition  of  the  integral  curvatures  which  correspond  to  the  single  parts. 
So,  generally,  the  integral  curvature  of  a  figure  is  equal  to  fJcdcr,  da  denoting  the 
element  of  area  of  the  figure,  and  h  the  measure  of  curvature  at  any  point.  The  prin- 
cipal points  concerning  the  geometric  representation  of  this  integral  reduce  to  the  fol- 
lowing. To  the  perimeter  of  the  figure  on  the  curved  surface  (under  the  restriction 
of  Art.  3)  will  correspond  always  a  closed  line  on  the  sphere.  If  the  latter  nowhere 
intersect  itself,  it  will  divide  the  whole  surface  of  the  sphere  into  two  parts,  one  of 
which  will  correspond  to  the  figure  on  the  curved  surface ;  and  its  area  (taken  as 
positive  or  negative  according  as,  with  respect  to  its  perimeter,  its  position  is  similar, 
or  inverse,  to  the  position  of  the  figure  on  the  curved  surface)  will  represent  the  inte- 
gral curvature  of  the  figure  on  the  curved  surface.  But  whenever  this  line  intersects 
itself  once  or  several  times,  it  will  give  a  complicated  figure,  to  which,  however,  it  is 
possible  to  assign  a  definite  area  as  legitimately  as  in  the  case  of  a  figure  without 
nodes ;  and  this  area,  properly  interpreted,  wiU  give  always  an  exact  value  for  the 
integral  curvature.  However,  we  must  reserve  for  another  occasion  the  more  extended 
exposition  of  the  theory  of  these  figures  viewed  from  this  very  general  standpoint. 

7. 

We  shall  now  find  a  formula  which  will  express  the  measure  of  curvature  for 
•  any  point  of  a  curved  surface.  Let  dcr  denote  the  area  of  an  element  of  this  surface ; 
p.f-  then  Zd(T  will  be  the  area  of  the  projection  of  this  element  on  the  plane  of  the  coor- 
dinates X,  y ;  and  consequently,  if  c?  S  is  the  area  of  the  corresponding  element  on  the 
sphere,  Z d'%  wiU  be  the  area  of  its  projection  on  the  same  plane.  The  positive  or 
negative  sign  of  Z  will,  in  fact,  indicate  that  the  position  of  the  projection  is  similar  or 
inverse  to  that  of  the  projected  element.  Evidently  these  projections  have  the  same 
ratio  as  to  quantity  and  the  same  relation  as  to  position  as  the  elements  themselves. 
Let  us  consider  now  a  triangular  element  on  the  curved  surface,  and  let  us  suppose 
that  the  coordinates  of  the  three  points  which  form  its  projection  are 

X,  y 

X  +  dx,    y  -^  dy 

X  +  S:r,    y  +  8y 


12  KARL  FRIEDRICH  GAUSS 

The  double  area  of  this  triangle  wiU  be  expressed  by  the  formula 

dx  .hi/  —  di/  .  Sa; 

and  this  will  be  in  a  positive  or  negative  form  according  as  the  position  of  the  side 
from  the  first  point  to  the  third,  with  respect  to  the  side  from  the  first  point  to  the 
second,  is  similar  or  opposite  to  the  position  of  the  ^-axis  of  coordinates  with  respect 
to  the  :r-axis  of  coordinates. 

In  like  manner,  if  the  coordinates  of  the  three  points  which  form  the  projection  of 
the  corresponding  element  on  the  sphere,  from  the  centre  of  the  sphere  as  origin,  are 

X,  Y 

X+dX,     Y+dY 
X+SX,     Y+SY 

the  double  area  of  this  projection  will  be  expressed  by 

dX.SY—dY.SX 

and  the  sign  of  this  expression  is  determined  in  the  same  manner  as  above.  Where- 
fore the  measure  of  curvature  at  this  point  of  the  curved  surface  will  be 

^       dX.SY—dY.SX 
dx  .  hy — di/ .  hx 

If  now  we  suppose  the  nature  of  the  curved  surface  to  be  defined  according  to  the  third 
method  considered  in  Art.  4,  X  and  Y  will  be  in  the  form  of  functions  of  the  quanti- 
ties X,  y.     We  shall  have,  therefore, 

^X  dX 

dX  —  ^ —  dx  +  -r-  dy 

ox  oy 

dx  dy     "^ 

dY  dY 

8F=  -::—bx-r  -ir-oy 
ox  Oy    '^ 

When  these  values  have  been  substituted,  the  above  expression  becomes 

_dX    BY       dX  ^dY 
dx      dy         dy      dx 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  13 

Setting,  as  above, 


and  also 


cs       ,     ds 

—  —  t,    —  =  M 

dx  dy 


dt  z=  Tdx  +  Udy,    du  =  Udx  +  V dy 


„PI     -(l-^ 


vc    ; 


and  hence 


we  have  from  the  formulae  given  above  .   -5  -"^     !•  ^^'^     '' 

X=-tZ,    Y^-uZ,    {l-Vf^-u^)Z^=\ 

dX^—Zdt—tdZ 

dY^=  — Zdu — u  dZ 

{l  +  f+u^)dZ+Z{tdt  +  udu)  =  0 

dZ=-Z'{tdt  +  udu) 

dX  =  -Z''{l  +  u^)  dt  +  ZHu  du 

dF=  +  Z'  tu  dt—Z'  (1  +  f)  du 

g  V" 

—  =  Z^i-{l  +  u^)T+tuUy 
~-=Z^i-{l  +  u')U+tuV^ 
^^Z^ituT-{l  +  f)U) 

^=Z^ituU-{l  +  f)Vy 

Substituting  these  values  in  the  above  expression,  it  becomes 

k^Z'{T  V—  m)  (1  +  ^+  u^)=Z'  [T  V—  U') 
TV—U' 


and  so 


{1  +  f+uY 


By  a  suitable  choice  of  origin  and  axes   of  coordinates,  we  can  easily  make  the 
values  of  the  quantities  t,  u,  U  vanish  for  a   definite  point  A.      Indeed,  the  first  two 


14  KARL  FRIEDEICH  GAUSS 

conditions  will  be  fulfilled  at  once  if  the  tangent  plane  at  this  point  be  taken  for  the 
a:^-plane.  If,  further,  the  origin  is  placed  at  the  point  A  itself,  the  expression  for 
the  coordinate  s  evidently  takes  the  form 

s=i  T°a?  +  U°xy  +  I  FY+  fi 

where  fl  will  be  of  higher  degree  than  the  second.  Turning  now  the  axes  of  x  and  p 
through  an  angle  M  such  that 

9  TT° 

tan2itf^  ^        .. 
T° F° 

it  is  easily  seen  that  there  must  result  an  equation  of  the  form 

z=k  Ta^+  hVtf+  D. 

In  this  way  the  third  condition  is  also  satisfied.  When  this  has  been  done,  it  is  evi- 
dent that 

I.  If  the  curved  surface  be  cut  by  a  plane  passing  through  the  normal  itself  and 
through  the  z-axis,  a  plane  curve  will  be  obtained,  the  radius  of  curvature  of  which 

at  the  point  A  wUl  be  equal  to  -„,  the  positive  or  negative  sign  indicating  that  the 

curve  is  concave  or  convex  toward  that  region  toward  which  the  coordinates  s  are 
positive. 

II.  In  like  manner  -^^  will  be  the  radius  of  curvature  at  the  point  A  of  the  plane 

curve  which  is  the  intersection  of  the  surface  and  the  plane  through  the  j/-axis  and 
the  s-axis. 

III.  Setting  x  —  r  GOS<p,  ?/  =  r  sin  <f),  the  equation  becomes 

s=i{T  cos^  <}>+V sin^  </>)  r^+  n 

from  which  we  see  that  if  the  section  is  made  by  a  plane  through  the  normal  at  A 
and  making  an  angle  <^  with  the  a;-axis,  we  shall  have  a  plane  curve  whose  radius  of 
curvature  at  the  point  A  wiU  be 

1 
rcos^<^+Fsin^<^ 

IV.  Therefore,  whenever  we  have  T—V,  the  radii  of  curvature  in  all  the  normal 
planes  will  be  equal.  But  if  T  and  V  are  not  equal,  it  is  evident  that,  since  for  any 
value  whatever  of  the  angle  </>,  T  cos^  (f)+  V  sin^  <^  falls  between  T  and  V,  the  radii  of 
curvature  in  the  principal  sections  considered  in  I.  and  II.  refer  to  the  extreme  curva- 
tures ;  that  is  to  say,  the  one  to  the  maximum  curvature,  the  other  to  the  minimum, 


GEKEEAL  EsTVESTIGATIONS  OF  CUEVED  SUKFACES  15 

if  T  and  F  have  the  same  sign.  On  the  other  hand,  one  has  the  greatest  convex 
curvature,  the  other  the  greatest  concave  curvature,  if  T  and  V  have  opposite  signs. 
These  conclusions  contain  almost  all  that  the  illustrious  Euler  was  the  first  to  prove 
on  the  curvature  of  curved  surfaces. 

V.  The  measure  of  curvature  at  the  point  A  on  the  curved  surface  takes  the 
very  simple  form 

k=TV, 
whence  we  have  the 

Theorem.  The  measure  of  curvature  at  any  point  whatever  of  the  surface  is  equal  to  a 
fraction  whose  numerator  is  unity,  and  whose  denominator  is  the  product  of  the  ttvo  extreme 
radii  of  curvature  of  the  sections  by  normal  planes. 

At  the  same  time  it  is  clear  that  the  measure  of  curvature  is  positive  for  con- 
cavo-concave or  convexo-convex  surfaces  (which  distinction  is  not  essential),  but  nega- 
tive for  concavo-convex  surfaces.  If  the  surface  consists  of  parts  of  each  kind,  then 
on  the  lines  separating  the  two  kinds  the  measure  of  curvature  ought  to  vanish.  Later 
we  shall  make  a  detailed  study  of  the  nature  of  curved  surfaces  for  which  the  meas- 
ure of  curvature  everywhere  vanishes. 

9. 

The  general  formula  for  the  measure  of  curvature  given  at  the  end  of  Art.  7  is 
the  most  simple  of  all,  since  it  involves  only  five  elements.  We  shall  arrive  at  a 
more  complicated  formula,  indeed,  one  involving  nine  elements,  if  we  wish  to  use  the 
first  method  of  representing  a  curved  surface.  Keeping  the  notation  of  Art.  4,  let  us 
set  also 


^!^=p,  ^^=0'  ^^-W 

a'^^^„     an^^^,,     a^^^„ 


5v 


dy  .dz  '    dx.ds       ^  '    dx.dy 

SO  that 

dP  =  P'  dx  +  R"  dy  +  Q"  ds 
dQ  =  R"dx+  Q!  dy  +P"dz 
dR  =  Q"  dx  +  P"dy  +R'  ds 

P 

Now  since  t  —  ~^i  we  find  through  difi'erentiation 

R'dt  =  -RdP  +  PdR  =  {PQ"—RP')dx  +  {PP"—RR")dy+{PR'—RQ")dz 


16  KAKL  FRIEDRICH  GAUSS 

or,  eliminating  dz  by  means  of  the  equation 

R^ dt={—R'' P' ^  2 P RQ"—P^ R')  dx  +{PRP"+  QRQ"—PQR'—R^R")dy. 

1  i   •  b  '^ 

In  like  manner  we  obtain  r 

R^du  =  {PRP"+QRQ"—PQR'—R'R")dx+{—R'Q'+2  QRP"—(^R')dy 

From  this  we  conclude  that 

WT=-R''P'^-'lPRq"—P^R' 
R^U=^PRP"-V  QRQ"-PQR'-R'R" 
R^V^-R^Q'+2  QRP"-Q^R' 

Substituting  these  values  in  the  formula  of  Art.  7,  we  obtain  for  the  measure  of  curv- 
ature k  the  following  symmetric  expression : 

(P^+  Q^+R-'fk=P^{Q'R'-P"^)+Q\P'R'-Q"^)  +R\P'Q'-R"^) 
+  2  QR{Q" R" -P' P")  +  2  PR {P" R" - Q' Q")  +  2PQ  {P"  Q" - R' R") 

10. 

r  We  obtain  a  still  more  complicated  formula,  indeed,  one  involving  fifteen  elements, 

if  we  follow  the  second  general  method  of  defining  the  nature  of  a  curved  surface.  It 
is,  however,  very  important  that  we  develop  this  formula  also.  Retaining  the  nota- 
tions of  Art.  4,  let  us  put  also 

■^  ^%         ■   ,        a/~^'    dp.dq    f^'    dq^     ^ 


and  let  us  put,  for  brevity,  «     y,-) 


Ic'-cV^A 
ca' — ac'===B 
aV-ba'  =  C 

First  we  see  that 

Adx^Bdy+Cd3==^, 

or 

a2  =  — Yfdx — jT  dy. 


O'^   -. 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  17 

Thus,  inasmuch  as  z  may  be  regarded  as  a  function  of  x,  y,  we  have 

dx^'^~       G 

d3_   ^_B_ 

dy-""-      C 
Then  from  the  formulae 

dx  —  adp-\- a' dq,    dy  —  bdp  +  b'dq,  (5  >  "  I  ■ 

we  have 

Cdp^=      V  dx  —  a'  dy 

Cdq'=—b  dx  +  a  dy 
Thence  we  obtain  for  the  total  differentials  of  ^,  m 

C^dt=lA^.-C^){b'dx-a'dy)  +  {0^-A^){bdx-ady) 
\       dp  dpi  \       d  q  cql 

C^du=^(B^-C^-p\{b'dx-a'dy)  +  (c^-^-B^){hdx-ady) 
\      dp  dp '  \      d  q  d  q  I 

If  now  we  substitute  in  these  formulae 

^-^-c'fi^by'-c^'-b'y 

dA 

-j^  =  c'fi'^by"-c^"-b'y' 

dB 

-z —  ='a'  y  +  e  a'  —  ay'  —c  a 

d  7? 

^-^a'y'+ea."-ay"-c'  a' 
dq  '  ' 

~  =  b'  a+afi'  -ba'  -a'  ^ 

^  =  b'a'+a  /8"-  b  a"-  a'  /8' 

and  if  we  note  that  the  values  of  the  differentials  dt,  du  thus  obtained  must  be  equal, 
independently  of  the  differentials  dx,  dy,  to  the  quantities  T dx-VU dy,  Udx+Vdy 
respectively,  we  shall  find,  after  some  sufficiently  obvious  transformations, 

C''T=aAb'^+/3Bb'^+yCb'^ 

-2  a'  Abb'-2  13'  Bbb'-2y'  Cbb' 
+  a"Ab'  +  ^"Bb''+y"Cb'' 


18  KARL  FRIEDRICH  GAUSS 

C^  i7=  -aAa'h'-^Ba'b'- y  Ca'V 

+  a'  il  {aV^  ha')+^'  B  [aV  +  ba')  +  y'  C{ab'+  ba') 

—  a"Aab-^"Bab-y"Cab 
C^V=aAa'^  +  ^Ba'^+yCa'^ 

—  2a'Aaa'—2^'Baa'—'iy'Caa' 
.    +0."  Aa^+  ^"  Ba^+y"  Ca"" 

Hence,  if  we  put,  for  the  sake  of  brevity, 

Aa   +BI3   +Cy  =D (1) 

Aa'  +B/3'  +  Cy'  =1)' (2) 

Aa"+BI3"+Cy"=-D" (3) 

we  shall  have 

C  T=D  V-  2  D'bb'  +  D"  b"- 

C^  U=-Da'b'  +  D'  {ab'+  ba')  -D"  ab 

Cn^=D  a'^-2  D'  aa'  +  D"  a" 

From  this  we  find,  after  the  reckoning  has  been  carried  out, 

C  {T  V-  m)  =  {DD"-D'^)  {ab'-ba'f  =  {DD"-D'^)  C 

and  therefore  the  formula  for  the  measure  of  curvature 

_    DD"—D"' 


11. 

By  means  of  the  formula  just  found  we  are  going  to  establish  another,  which  may 
be  counted  among  the  most  productive  theorems  in  the  theory  of  curved  surfaces. 
Let  us  introduce  the  following  notation  : 

■>%'^>,"  aa'+bb'+cc'=F 

•yl  a'^+b'^^c'^=G 

^^'  V?4     (i  o.  +b  ^  +cy  =m (4) 

\   ~^f    a  a'  ^-b  fi'  +c  y'  =m' (5) 

-BF       .1^"  a  a"+b  fi"^-c  y"  =  m" (6) 

Vi'    '"^T  a'a  -^b'  fi   +c'y  =n (7) 

f^,_aJl>p^'   a'a'^b'^'+c'y-=n' (8) 

'        u'a"+b'  fi"+c'y"^n" (9) 

A^-\-  B'-V  C^=EG—F^=^ 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  19 

Let  us  eliminate  from  the  equations  1,  4,  7  the  quantities  13,  y,  which  is  done  by 
multiplying  them  by  bc'—cb',  h'C—c'B,  c -5  —  J  C  respectively  and  adding.  In  this 
way  we  obtain 

(A{hc'-cb')  +  a{b'C-c'B)  +  a'{cB-bC))a 
=  D{bc'—cb')  +  m{b'  C-  c'B)  +  n {cB  -  b  C) 

an  equation  which  is  easily  transformed  into 

AD  =  ai^  +  a{nF—mG)  +  a'{mF~nE) 

Likewise  the  elimination  of  a,  y  or  a,  /8  from  the  same  equations  gives 

BD=-^^  +  b{nF-mG)  +  b'  {niF  —  nE) 
CD  =  y^+  c{nF  —  mG)  +  c'  {mF  —  nE) 

Multiplying  these  three  equations  by  a",  /8",  y"  respectively  and  adding,  we  obtain 

DD"={aa"+  ^fi"+yy")^+m"{nF-mG)+n"  (mF-nE)     .     .     .     (10) 

If  we  treat  the  equations  2,  5,  8  in  the  same  way,  we  obtain 

AD'=  a'  A+  a  (n'F-m'G)  +  a'  [m'F-n'E) 
BD'=  ^'A+b  {n'F-  m' G)  +  b'  {m'F-n'E) 
CD'  =  y'  £^+  c  {n'  F  -m'  G)  +  c'  {m'F-n'E) 

and  after  these  equations  are  multiplied  by  a',  /8',  y'  respectively,  addition  gives 

D'^=  (a''+  /8''+  r")  A  +  m'  {n'F-m'  G)  +  n' {m'F-n'E) 

A  combination  of  this  equation  with  equation  (10)  gives 

DD"-D'^={aa"+^^"+yy"-a'^-fi'^-y'^)A 

+  E{n'^-nn")  +F{nm"-  2  m'n'+mn")  +  G{m'^-mm") 
It  is  clear  that  we  have 

3^  ' 


dE 

dE 

dF 

dF 

dG 

=  2m, 

=  2m' 

=  m'  + 

n, 

^    =m 

"  +  n' 

=  2< 

dp 

dq 

'      dp 

,  dE 

dq 
,  dE 

'      dp 

dF 

,  dG 

tn  = 

dp 

dF 

,  dE 

m' 

dq 

,dG 

m"  = 

dq 

^dG 

dp 

n  = 

dp 

dq 

n'- 

^  dp' 

n"  = 

*a. 

Moreover,  it  is  easily  shown  that  we  shall  have 

aa"  +  l3l3"  +  yy"-a'^-(3''-y'^=j^~^  =  -jy-j^ 

=  _i  g'-g     3'F       d'G 

"  dq^       dp.dq  '  dp^ 


'^yr 


\...-s,,ir^^''^^" 


k>  - 


%J 


20 


KARL  FRIEDIilCH  GAUSS 


^'t 


If  we  substitute  these  different  expressions  in  the  formula  for  the  measure  of  curva- 
ture derived  at  the  end  of  the  preceding  article,  we  obtain  the  following  formula,  which 
involves  only  the  quantities  U,  F,  G  and  their  differential  quotients  of  the  first  and 
second  orders : 

dE     SG__^dF_    dG  .  idG\ 

dq  ■  ■  —  ■' 

dp  dq       dq 


i{I!G-F'fk=Js(^ 

a£_a^    d_G 

dq         dq 


fdE 
dp 

•dE 

dp 


dq 
dE 


dp       dq        \  dp  '  / 


dE         dE 

dp 


dE 

dq 


dE 


dp 


dG       dE 

dp  dp 


dE 

dq 


,(m')-.i.o^nm-.^yS 


d'E 


d_G_ 

dp 

d'^E 


dp  .dq       dp 


12. 


Since  we  always  have 

d^+  df+  dz^  =  Edp''+2  Edp  .dq+G dq\ 
it  is  clear  that 

V{Ed2f+2Edp.dq+Gdq^) 

is  the  general  expression  for  the  linear  element  on  the  curved  surface.  The  analysis 
developed  in  the  preceding  article  thus  shows  us  that  for  finding  the  measure  of  cur- 
vature there  is  no  need  of  finite  formulae,  which  express  the  coordinates  x,  y,  z  as 
functions  of  the  indeterminates  p,  q  ;  but  that  the  general  expression  for  the  magnitude 
of  any  linear  element  is  sufficient.  Let  us  proceed  to  some  applications  of  this  very 
important  theorem. 

Suppose  that  our  surface  can  be  developed  upon  another  surface,  curved  or  plane, 
so  that  to  each  point  of  the  former  surface,  determined  by  the  coordinates  x,  y,  z,  will 
correspond  a  definite  point  of  the  latter  surface,  whose  coordinates  are  x',  y' ,  s' .  Evi- 
dently a;',  y' ,  z'  can  also  be  regarded  as  functions  of  the  indeterminates  p,  q,  and  there- 
fore for  the  element  V  {dx''^+  dy''^-\-  dz''^)  we  shall  have  an  expression  of  the  form 

\/{E'dp^+  2E'dp.dq  +  G'dq^) 

where  E' ,  E' ,  G'  also  denote  functions  of  p,  q.  But  from  the  very  notion  of  the  devel- 
opment of  one  surface  upon  another  it  is  clear  that  the  elements  corresponding  to  one 
another  on  the  two  surfaces  are  necessarily  equal.     Therefore  we  shall  have  identically 

E^E',   E^E',    G^G'. 

Thus  the  formula  of  the  preceding  article  leads  of  itself  to  the  remarkable 

TuEOREM.  If  a  curved  surface  is  developed  upon  any  other  surface  whatever,  the 
measure  of  curvature  in  each  point  remains  unchanged. 


GEISTEEAL  INVESTIGATIONS  OF  CUKVED  SURFACES 


21 


Also  it  is  evident  that  any  finite  part  whatever  of  the  curved  surface  loill  retain  the 
same  integral  curvature  after  development  upon  another  surface. 

Surfaces  developable  upon  a  plane  constitute  the  particular  case  to  which  geom- 
eters have  heretofore  restricted  their  attention.  Our  theory  shows  at  once  that  the 
measure  of  curvature  at  every  point  of  such  surfaces  is  equal  to  zero.  Consequently, 
if  the  nature  of  these  surfaces  is  defined  according  to  the  thu-d  method,  we  shall  have 
at  every  point 

d'^Z       d'^Z  I    d^z     \''_ 

a  criterion  which,  though  indeed  known  a  short  time  ago,  has  not,  at  least  to  our 
knowledge,  commonly  been  demonstrated  with  as  much  rigor  as  is  desirable."^ 

13. 

What  we  have  explained  in  the  preceding  article  is  connected  with  a  particular 
method  of  studying  surfaces,  a  very  worthy  method  which  may  be  thoroughly  devel- 
oped by  geometers.  When  a  surface  is  regarded,  not  as  the  boundary  of  a  solid,  but 
as  a  flexible,  though  not  extensible  solid,  one  dimension  of  which  is  supposed  to 
vanish,  then  the  properties  of  the  surface  depend  in  part  upon  the  form  to  which  we 
can  suppose  it  reduced,  and  in  part  are  absolute  and  remain  invariable,  whatever  may 
be  the  form  into  which  the  surface  is  bent.  To  these  latter  properties,  the  study  of 
which  opens  to  geometry  a  new  and  fertile  field,  belong  the  measure  of  curvature  and 
the  integral  curvature,  in  the  sense  which  we  have  given  to  these  expressions.  To 
these  belong  also  the  theory  of  shortest  lines,  and  a  great  part  of  what  we  reserve  to 
be  treated  later.  From  this  point  of  view,  a  plane  surface  and  a  surface  developable 
on  a  plane,  e.  g.,  cylindrical  surfaces,  conical  surfaces,  etc.,  are  to  be  regarded  as  essen- 
tially identical;  and  the  generic  method  of  defining  in  a  general  manner  the  nature  of 
the  surfaces  thus  considered  is  always  based  upon  the  formula 

V{Edp''-\-  2  Fdp  .dq^a  dq% 

which  connects  the  linear  element  with  the  two  indeterminates  p,  q.  But  before  fol- 
lowing this  study  further,  we  must  introduce  the  principles  of  the  theory  of  shortest 
lines  on  a  given  curved  surface. 

14. 

The  nature  of  a  curved  line  in  space  is  generally  given  in  such  a  way  that  the 
coordinates  x,  y,  z  corresponding  to  the  diflferent  points  of  it  are  given  in  the  form  of 
functions  of  a  single  variable,  which  we  shall  call  w.     The  length  of  such  a  line  from 


^  fw-^,r\(\[^ 


y 


/ 


4/\  »^v«*i-^.^j^- 


A  ^.^-^ 


22  KARL  FRIEDRICH  GAUSS 

an  arbitrary  initial  point  to  the  point  whose  coordinates  are  x,  y,  z,  is   expressed  ))y 
the  integral  \    iV"-^ 


f'-Mthm-m)    ^-^''^/' 


If  we  suppose  that  the  position  of  the  line  undei'goes  an  infinitely  small  variation,  so 
that  the  coordinates  of  the  different  points  receive  the  variations  hx,  Sy,  8  s,  the  varia- 
tion of  the  whole  length  becomes 

'dx  .dSx  +dy  .dSt/  +  ds  .dSs 


r 


Vidx'+df+ds') 
which  expression  we  can  change  into  the  form 

dx  .8x-\-d7/  .hy  -\-ds  .  8z 

vJdJ+dfTdJ) 

J  V^-^Vidx'+df+dz')  '^^^•^V{dx'+df+d^)  '^^^•'^  Vidx'+df+dzy 

We  know  that,  in  case  the  line  is  to  be  the  shortest  between  its  end  points,  all  that 
stands  under  the  integral  sign  must  vanish.  Since  the  line  must  Ue  on  the  given 
surface,  whose  nature  is  defined  by  the  equation 

Pdx  +  Qdy+Rds-=0, 
the  variations  Sx,  By,  8s  also  must  satisfy  the  equation 

PSx  +  Q8y+RSs  =  0, 
and  from  this  it  follows  at  once,  according  to  well-known  rules,  that  the  differentials 

d  _  , ,  n  .1  1    J  n  I    7  „2 \  >    d  _ ,  I J  ^1,  I    7.9  I    T72V'    d- 


V{d:^-^dy'\d^Y    "^  V  {dx"^  dy''+ ds^Y    'W {d:^^  dy''-^  ds^) 

must  be  proportional  to  the  quantities  P,  Q,  R  respectively.  Let  c?;-  be  the  element 
of  the  curved  line ;  X  the  point  on  the  sphere  representing  the  direction  of  this  ele- 
ment ;  L  the  point  on  the  sphere  representing  the  direction  of  the  normal  to  the  curved 
surface ;  finally,  let  ^,  -q,  t,  be  the  coordinates  of  the  point  \,  and  X,  Y,  Z  be  those  of 
the  point  L  with  reference  to  the  centre  of  the  sphere.  We  shall  then  have 
dx~$dr,    dy  =  r}dr,    ds='l,dr 

from  which  we  see  that  the  above  differentials  become  d^,  d-q,  dt,.  And  since  the 
quantities  P,  Q,  R  are  proportional  to  X,  Y,  Z,  the  character  of  shortest  lines  is 
expressed  by  the  equations 

dl^drj  ^dC  ^  > 

X"  Y  ~  Z  o" 


GENERAL  INVESTIGATIONS  OF  CUEVED  SITRFACES  23 

Moreover,  it  is  easily  seen  that 

is  equal  to  the  small  arc  on  the  sphere  which  measures  the  angle  between  the  direc- 
tions of  the  tangents  at  the  beginning  and  at  the  end  of  the  element  dr,  and  is  thus 

equal  to  — ?  if  p  denotes  the  radius   of  curvature  of  the  shortest  line  at  this   point. 
Thus  we  shall  have 

pd^=Xdr,    pdrj  =  Ydr,    pdt,^Zdr 

15. 

Suppose  that  an  infinite  number  of  shortest  lines  go  out  from  a  given  point  A 
on  the  curved  surface,  and  suppose  that  we  distinguish  these  lines  from  one  another 
by  the  angle  that  the  first  element  of  each  of  them  makes  with  the  first  element  of 
one  of  them  which  we  take  for  the  first.  Let  j>  be  that  angle,  or,  more  generally,  a 
function  of  that  angle,  and  ?•  the  length  of  such  a  shortest  line  from  the  point  A  to 
the  point  whose  coordinates  are  z,  y,  z.  Since  to  definite  values  of  the  variables  r,  <^ 
there  correspond  definite  points  of  the  surface,  the  coordinates  x,  y,  z  can  be  regarded 
as  functions  of  r,  ^.  We  shall  retain  for  the  notation  A,  L,  ^,  rj,  ^,  X,  Y,  Z  the  same 
meaning  as  in  the  preceding  article,  this  notation  referring  to  any  point  whatever  on 
any  one  of  the  shortest  lines. 

All  the  shortest  lines  that  are  of  the  same  length  r  wUl  end  on  another  line 
whose  length,  measured  from  an  arbitrary  initial  point,  we  shall  denote  by  v.  Thus  v 
can  be  regarded  as  a-  function  of  the  indeterminates  r,  <f>,  and  if  X'  denotes  the  point 
on  the  sphere  corresponding  to  the  direction  of  the  element  dv,  and  also  ^',  tj,'  ^' 
denote  the  coordinates  of  this  point  with  reference  to  the  centre  of  the  sphere,  we 
shall  have 

a^^^'a^'   d^~"^'d^'    d^^^'d^ 

From  these  equations  and  from  the  equations 

a^_      ciy  _      ^— y 

dr~^'    dr~'^'    dr^^ 
we  have 

dx     dx      dy     dy      dz     dz       ,.^,,         ,  j_  r  ri\    ^^  \\/    ^^ 

.  a7-a^  +  a7 -3^  +  97  •a^=^(^^+^''+^^) 'a^^^^^^^a^ 


24  KARL  FRIEDRICH  GAUSS 

Let  S  denote  the  first  member  of  this  equation,  which  will  also  be  a  function  of  r,  <^. 
Diflferentiation  of  S  with  respect  to  r  gives 


d?'    d(j)       dr    d^        dr    dif)  d<f> 


But 


and  therefore  its   differential  is  equal  to  zero  ;   and  by  the  preceding  article  we  have, 
if  p  denotes  the  radius  of  curvature  of  the  line  r, 

3l_x    a^_r    H__z_ 

dr       p'     dr       p '     dr       p 
Thus  we  have 

^=  1 .  (Xf  +  Y-q'  +ZC')  .  ^  =  1 .  cos  iX' .  ^  =  0 
dr       p  3(p      p  d(p 

since   X'  evidently  lies   on  the  great  circle  whose  pole  is  L.     From  this  we  see  that 
>S'  is  independent  of  r,  and  is,  therefore,  a  function  of  <f)  alone.     But  for  r  =  0  we  evi- 

dv 

dently  have  v  —  0,  consequently  ^  —  Oj  and  >S'=  0  independently  of  tf).     Thus,  in  general, 

we  have  necessarily /S'=  0,  and  so  cos\\'=0,i.e.,  XX'=90°.      From  this  follows  the 
Theorem.     If  on  a  curved  surface  an  infinite  number  of  shortest  lines  of  equal  length 
he  dravm  from  the  same  initial  point,  the  lines  joining  their  extremities  will  le  normal  to 
each  of  the  lines. 

We  have  thought  it  worth  while  to  deduce  this  theorem  from  the  fundamental 
property  of  shortest  lines  ;  but  the  truth  of  the  theorem  can  be  made  apparent  with- 
out any  calculation  by  means  of  the  following  reasoning.  Let  AB,  AB'  be  two 
shortest  lines  of  the  same  length  including  at  A  an  infinitely  small  angle,  and  let  us 
suppose  that  one  of  the  angles  made  by  the  element  B  B'  with  the  lines  B  A,  B'  A 
differs  from  a  right  angle  by  a  finite  quantity.  Then,  by  the  law  of  continuity,  one 
wiU  be  greater  and  the  other  less  than  a  right  angle.  Suppose  the  angle  at  B  is 
equal  to  90° — w,  and  take  on  the  line  AB  ?i  point  C,  such  that 

BC=BB'.  cosec  &>. 

Then,  since  the  infinitely  small  triangle  BB'Cmsiy  be  regarded  as  plane,  we  shall  have 

CB'  =  BC  .cos  01, 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  25 

and  consequently 

AC-^  CB'^AC^BC  .  co&ca^AB—BC  .  {l-costo)=AB'-B C  .  (1-cosw), 

i.  e.,  the  path  from  A  to  B'  through  the  point  0  is  shorter  than  the  shortest  line, 
Q.  E.  A. 

16. 

With  the  theorem  of  the  preceding  article  we  associate  another,  which  we  state 
as  follows  :  If  on  a  curved  surface  we  imagine  any  line  tvhatever ,  from  the  different  points 
of  which  are  drawn  at  right  angles  and  toward  the  same  side  an  infinite  number  of  shortest 
lines  of  the  same  length,  the  curve  which  joins  their  other  extremities  vnll  cut  each  of  the 
lines  at  right  angles.  For  the  demonstration  of  this  theorem  no  change  need  be  made 
in  the  preceding  analysis,  except  that  <^  must  denote  the  length  of  the  given  curve 
measured  from  an  arbitrary  point ;  or  rather,  a  function  of  this  length.  Thus  all  of 
the  reasoning  will  hold  here  also,  with  this  modification,  that  ^S^O  for  ;*  =  0  is 
now  implied  in  the  hypothesis  itself.  Moreover,  this  theorem  is  more  general  than 
the  preceding  one,  for  we  can  regard  it  as  including  the  first  one  if  we  take  for  the 
given  line  the  infinitely  small  circle  described  about  the  centre  A.  Finally,  we  may 
say  that  here  also  geometric  considerations  may  take  the  place  of  the  analysis,  which, 
however,  we  shall  not  take  the  time  to  consider  here,  since  they  are  sufficiently 
obvious. 

17. 
We  return  to  the  formula 

V{E  df^  2  Fdf  .dq  +  a  df\ 

which  expresses  generally  the  magnitude  of  a  linear  element  on  the  curved  surface, 
and  investigate,  first  of  all,  the  geometric  meaning  of  the  coefficients  E,  F,  G.  We 
have  already  said  in  Art.  5  that  two  systems  of  lines  may  be  supposed  to  lie  on  the 
curved  surface,  p  being  variable,  q  constant  along  each  of  the  lines  of  the  one  system ; 
and  q  variable,  p  constant  along  each  of  the  lines  of  the  other  system.  Any  point 
whatever  on  the  surface  can  be  regarded  as  the  intersection  of  a  line  of  the  first 
system  with  a  line  of  the  second  ;  and  then  the  element  of  the  first  line  adjacent  to 
this  point  and  corresponding  to  a  variation  dp  will  be  equal  to  V E .  dp,  and  the 
element  of  the  second  line  corresponding  to  the  variation  dq  wlU  be  equal  to  i/  (7 .  dq. 
Finally,  denoting  by  w  the  angle  between  these  elements,  it  is  easUy  seen  that  we 
shall  have 

F 

^^^^  =  Veg- 


^- 


■Iq  Vf 


M^' 


26  KAEL  FRIEDRICH  GAUSS 

Furthermore,  the  area  of  the  surface  element  in  the  form  of  a  parallelogram  between 
the  two  lines  of  the  first  system,  to  which  correspond  q,  q-\-  dq,  and  the  two  lines  of 
the  second  system,  to  which  correspond  p,  p  -^  dp,  will  be 

V{Ea-F^)dp.dq. 

Any  line  whatever  on  the  curved  surface  belonging  to  neither  of  the  two  sys- 
tems is  determined  when  p  and  q  are  supposed  to  be  functions  of  a  new  variable,  or 
one  of  them  is  supposed  to  be  a  function  of  the  other.  Let  s  be  the  length  of  such 
a  curve,  measured  from  an  arbitrary  initial  point,  and  in  either  direction  chosen  as 
positive.     Let  6  denote  the  angle  which  the  element 

ds=^V{Edp^\  2  Fdp  .  dq  +  Gdf) 

makes  with  the  line  of  the  first  system  drawn  through  the  initial  point  of  the  ele- 
ment, and,  in  order  that  no  ambiguity  may  ai'ise,  let  us  suppose  that  this  angle  is 
measured  from  that  branch  of  the  first  line  on  which  the  values  of  p  increase,  and  is 
taken  as  positive  toward  that  side  toward  which  the  values  of  q  increase.  These  con- 
ventions being  made,  it  is  easUy  seen  that 

cos  d  .  ds  =  V E .  dp  +  V  G  .  cos  co  .  dq  = ^  ,  „ — ^ 

V  E 

■     a     -,  yn      ■  J         v'iEG-F^).dq 

sm  0  .ds  =  y  G  .sm  0)  .dq  =  — !^ ' i 

V  E 

18. 

We  shall  now  investigate  the  condition  that  this  line  be  a  shortest  line.  Since 
its  length  s  is  expressed  by  the  integral 

s  =J  V{Edp^^  2  Fdp  .dq^G  dq^) 

the  condition  for  a  minimum  requires  that  the  variation  of  this  integral  arising  from 
an  infinitely  small  change  in  the  position  become  equal  to  zero.  The  calculation,  for 
our  purpose,  is  more  simply  made  in  this  case,  if  we  regard  ji;  as  a  function  of  q. 
When  this  is  done,  if  the  variation  is  denoted  by  the  characteristic  S,  we  have 

//^  .dp^Jr^  —  'dp-dq-^—--  dq-'X  Sp  +  (2  Edp  +  2  Fdq)  d8p 
_dp Sp a|_ 

_Edp+Fdq    g 


GENERAL  INTESTIGATIONS  OF  CURVED  SURFACES  27 

dE     ,  „      ^dF     ,       ,        dG 


+    l^^^(^Sp-'^^''  +  ^3p-'^f-'^^+3p-'^^\    ,   Udp+Fdg 


2d} 


and  we  know  that  what  is  included  under  the  integral  sign  must  vanish  independently 
of  8^.     Thus  we  have 

l^.df+2'-^.dp.d,  +  ^.df=2ds.d.l^I^+Zj^ 

dp     ^  ^    dp     ^     ^      dp     ^  ds 

=  2  ds.d.VE  .Gosd 

=  ^ll^l^^^ -2  ds.d0.VE. sine 

VE  j  j 

■    ^iE^P±Fdq)j^_^^Ea-F^).d^^.de  "^1         f/ 

=  (^±+l^).^-^^.dp+'^.dq)-2v{EG-F^).dg.dd 
This  gives  the  following  conditional  equation  for  a  shortest  line  : 

y  /  n  ^         n  9\       in         1      F       dE       7  1.      F       dE       ^  1       dE       7 

y{^G-^')-dO  =  ^.-.  —  .dp  +  -.-.  —  .dg+^.  —  .dp 

dF     ■,        1    dG     . 
-^•^P-2^-^^ 
which  can  also  be  written 


From  this  equation,  by  means  of  the  equation 

E dp F 


cot^^-— ^^_.'^  + 


^ 


V{EG-F^)   dq  '   V{EG-F^) 

it  is  also  possible  to  eliminate  the  angle  6,  and  to  derive  a  differential  equation  of 
the  second  order  between  p  and  q,  which,  however,  would  become  more  complicated 
and  less  useful  for  applications  than  the  preceding. 

19. 

The  general  formulae,  which  we  have  derived  in  Arts.  11,  18  for  the  measure  of 
curvature  and  the  variation  in  the  direction  of  a  shortest  line,  become  much  simpler 
if  the  quantities  p,  q  are  so  chosen  that  the  lines  of  the  first  system  cut  everywhere 


28  KARL  FRIEDRICH  GAUSS 


orthogonally  the  lines  of  the  second  system;  i.  e.,  in  such  a  way  that  we  have  gen- 
erally w  =  90°,  or  ^=0.     Then  the  formula  for  the  measure  of  curvature  becomes 

dq      dq  \dpf^^     dp      dp^^Xdqf  Xdq^^dp^r 

and  for  the  variation  of  the  angle  6 

VEG.de  =  l.^.dp-l.^.dq 
2     dq       ^       ^     dp        ^ 

Among  the  various  cases  in  which  we  have  this   condition  of  orthogonality,  the 

most  important  is  that  in  which  all  the  lines  of  one  of  the  two   systems,  e.  g.,  the 

first,  are  shortest  lines.     Here  for  a  constant  value  of  q  the  angle  Q  becomes  equal  to 

zero,  and  therefore  the  equation  for  the  variation  of  6  just  given  shows  that  we  must 

.  dE  .  .  \, 

Jiave  —==0,  or  that  the   coefficient  E  must  be  independent  of  «;    i.  e.,  E  must  be 

V  either  a  constant  or  a  function  of  p  alone.  It  will  be  simplest  to  take  for  p 
the  length  of  each  line  of  the  first  system,  which  length,  when  all  the  lines  of  the 
first  system  meet  in  a  point,  is  to  be  measured  from  this  point,  or,  if  there  is  no 
common  intersection,  from  any  line  whatever  of  the  second  system.  Having  made 
these  conventions,  it  is  evident  that  p  and  q  denote  now  the  same  quantities  that 
<*  were  expressed  in  Arts.  15,  16  by  r  and  <f),  and  that  ^=1.  Thus  the  two  preced- 
jjx^       ^  -C'  ^■K'^  ">    ^^S  formulae  become  : 

'^^a)  \dpf  dp^ 


s 


or,  setting   i/  G^=m, 


V'G.de  =  -l~.dq 
2    dp       ^ 

J              \     d'^m       J  a            dm     J 
k=- 2'     «"= -dq 


dp^  dp 

Generally  speaking,  ot  will  be  a  function  of  jo,  q,  and  mdq  the  expression  for  the  ele- 
ment of  any  line  whatever  of  the  second  system.  But  in  the  particular  case  where 
all  the  lines  p  go  out  from  the  same  point,  evidently  we  must  have  m  =  0  for  jo  =  0. 
Furthermore,  in  the  case  under  discussion  we  will  take  for  q  the  angle  itself  which 
the  first  element  of  any  line  whatever  of  the  first  system  makes  with  the.  element  of 
any  one  of  the  lines  chosen  arbitrarily.  Then,  since  for  an  infinitely  small  value  of 
p  the  element  of  a  line  of  the  second  system  (which  can  be  regarded  as  a  circle 
described  with  radius  j^)  is   equal  to  pdq,  we  shall  have  for  an  infinitely  small  value 

0?  p,m  =  p,  and  consequently,  for^  =  0,  m  =  0  at  the  same  time,  and   =j— ==1. 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  29 

20. 

We  pause  to  investigate  the  case  in  which  we  suppose  that  p  denotes  in  a  gen- 
eral manner  the  length  of  the  shortest  line  drawn  from  a  fixed  point  A  to  any  other 
point  whatever  of  the  surface,  and  q  the  angle  that  the  first  element  of  this  line 
makes  with  the  first  element  of  another  given  shortest  line  going  out  from  A.  Let 
5  be  a  definite  point  in  the  latter  line,  for  which  §'  =  0,  and  C  another  definite  point 
of  the  surface,  at  which  we  denote  the  value  of  q  simply  by  A.  Let  us  suppose  the 
points  B,  C  joined  by  a  shortest  Une,  the  parts  of  which,  measured  from  B,  we  denote 
in  a  general  way,  as  in  Art.  18,  by  s ;  and,  as  in  the  same  article,  let  us  denote  by  B 
the  angle  which  any  element  ds  makes  with  the  element  dp;  finally,  let  us  denote 
by  6°,  6'  the  values  of  the  angle  6  at  the  points  B,  C.  We  have  thus  on  the  curved 
surface  a  triangle  formed  by  shortest  lines.  The  angles  of  this  triangle  at  B  and  C 
we  shall  denote  simply  by  the  same  letters,  and  B  will  be  equal  to  180° — 9,  C  to  6' 
itself.  But,  since  it  is  easily  seen  from  our  analysis  that  aU  the  angles  are  supposed 
to  be  expressed,  not  in  degrees,  but  by  numbers,  in  such  a  way  that  the  angle  57°  17' 
45",  to  which  corresponds  an  arc  equal  to  the  radius,  is  taken  for  the  unit,  we  must  set 

e°  =  n—B,     d'=C 

where  2^7  denotes  the  circumference  of  the  sphere.     Let  us  now  examine  the  integral 
curvature  of  this  triangle,  which  is  equal  to 


J  k  da, 


da-  denoting  a  surface  element  of  the  triangle.     Wherefore,  since  this   element  is  ex- 
pressed by  m  dp  .  dq,  we  must  extend  the  integral 


ffmdp.dq 


over  the  whole  surface  of  the  triangle.      Let  us  begin  by  integration  with  respect  to 
p,  which,  because 

,  1     d^m 

gives 

dq.  (const. -^), 

for  the  integral  curvature  of  the  area  lying  between  the  lines  of  the  first  system,  to 
which  correspond  the  values  q,  q  +  dq  of  the  second  indeterminate.      Since  this  inte- 


k 


30  KARL  FRIEDRICH  GAUSS 

gral  curvature  must  vanish  for  p  =  0,  the  constant  introduced  by  integration  must  be 

equal  to  the  value  of  ^  for  p  =  0,  i.  e.,  equal  to  unity.      Thus  we  have 


dq[^ 


ap  I 


where  for  -^  must  be  taken  the  value  corresponding  to  the  end  of  this  area  on  the 
line  CB.     But  on  this  line  we  have,  by  the  preceding  article, 

dq        ^ 

whence  our  expression  is  changed  into  dq  +  dd.  Now  by  a  second  integration,  taken 
from  ^  =  0  to  ^^  =  ^,  we  obtain  for  the  integral  curvature 

A  +  d'-  e°, 

or 

A+B  +  C-TT. 

The  integral  curvature  is  equal  to  the  area  of  that  part  of  the  sphere  which  cor- 
responds to  the  triangle,  taken  with  the  positive  or  negative  sign  according  as  the 
curved  surface  on  which  the  triangle  lies  is  concavo-concave  or  concavo-convex.  For 
unit  area  will  be  taken  the  square  whose  side  is  equal  to  unity  (the  radius  of  the 
sphere),  and  then  the  whole  surface  of  the  sphere  becomes  equal  to  4  tt.  Thus  the 
part  of  the  surface  of  the  sphere  corresponding  to  the  triangle  is  to  the  whole  surface 
of  the  sphere  as  ±  ( J.  +  5  +  C  —  tt)  is  to  4  tt.  This  theorem,  which,  if  we  mistake 
not,  ought  to  be  counted  among  the  most  elegant  in  the  theory  of  curved  surfaces, 
may  also  be  stated  as  follows  : 

The  excess  over  180°  of  the  sum  of  the  angles  of  a  triangle  formed  by  showiest  lines 

■  \jyr^^-'     ^^  ^  concavo-concave  curved  surface,  or  the  deficit  from  180°  of  the  sum  of  the  angles  of 

{'  '  a  triangle  formed  hy  shortest  lines  on  a  concavo-convex  curved  surface,  is  measured  by  the 

'^  n»  \%%  "'"^*  '^f  ^^^  P*^^^  ^f  ^^^  sphere  tvhich  corresponds,  through  the  directions  of  the  normals,  to 

^  ^^  '^Vti.  that  triangle,  if  the  whole  surface  of  the  sphere  is  set  equal  to  720  degrees. 

'i\j^^f^  More  generally,  in  any  polygon  whatever  of  n  sides,  each  formed  by  a  shortest 

I U  *?  '     line,  the  excess  of  the  sum  of  the  angles  over  (2  n  —  4)  right  angles,  or  the  deficit  from 
'  (2 « —  4)  right  angles   (according  to  the  nature  of  the  curved  surface),  is  equal  to  the 

area  of  the  corresponding  polygon  on  the  sphere,  if  the  whole  surface  of  the  sphere  is 
set  equal  to  720  degrees.  This  follows  at  once  from  the  preceding  theorem  by  divid- 
ing the  polygon  into  triangles. 


GEKERAL  INVESTIGATIONS  OF  CURVED  SURFACES  31 

21. 

Let  us  again  give  to  the  symbols  p,  q,  E,  F,  G,  a>  the  general  meanings  which 
were  given  to  them  above,  and  let  us  further  suppose  that  the  nature  of  the  curved 
surface  is  defined  in  a  similar  way  by  two  other  variables,  p',  q',  in  which  case  the 
general  linear  element  is  expressed  by 

V{E'  dp'^+  2  F'  dp'.  dq'+G'  dq"') 

Thus  to  any  point  whatever  lying  on  the  surface  and  defined  by  definite  values  of 
the  variables  p,  q  wUl  correspond  definite  values  of  the  variables  p',  q',  which  will 
therefore  be  functions  of  p,  q.     Let  us  suppose  we  obtain  by  differentiating  them 

dp'—  adp+  /Bdq 
dq'=  y  d])  +  8  dq 

We  shaU  now  investigate  the  geometric  meaning  of  the  coefficients  a,  fi,  y,  8. 

Now  fom-  systems  of  lines  may  thus  be  supposed  to  lie  upon  the  curved  surface, 
for  which  p,  q,  p',  q'  respectively  are  constants.  If  through  the  definite  point  to 
which  correspond  the  values  p,  q,  p',  q'  of  the  variables  we  suppose  the  four  lines 
belonging  to  these  different  systems  to  be  drawn,  the  elements  of  these  Unes,  corres- 
ponding to  the  positive  increments  dp,  dq,  dp',  dq',  will  be 

VE.dp,     VG.dq,     VE'.dp',     VG'.dq'. 

The  angles  which  the  directions  of  these  elements  make  with  an  arbitrary  fixed  direc- 
tion we  shall  denote  by  M,  N,  M ,  N',  measuring  them  in  the  sense  in  which  the 
second  is  placed  with  respect  to  the  first,  so  that  sin(iV— Jlf)  is  positive.  Let  us 
suppose  (which  is  permissible)  that  the  fourth  is  placed  in  the  same  sense  with  respect 
to  the  third,  so  that  sin(iV''  —  M)  also  is  positive.  Having  made  these  conventions, 
if  we  consider  another  point  at  an  infinitely  small  distance  from  the  first  point,  and 
to  which  correspond  the  values  jo  +  rfjo,  q-\-  dq,  p'-{-  dp',  q'-\-dq'  of  the  variables,  we 
see  without  much  difficulty  that  we  shall  have  generally,  i.  e.,  independently  of  the 
values  of  the  increments  dp,  dq,  dp/,  dq', 

VE  .dp.  s,mM+VG  .  dq  .  sini\^=  VE'.  dp',  sinif  +  VG'.  dq'.  siniV 

since  each  of  these  expressions  is  merely  the  distance  of  the  new  point  from  the  line 
from  which  the  angles  of  the  directions  begin.  But  we  have,  by  the  notation  intro- 
duced above, 

N-M=o>. 
In  like  manner  we  set 

N'-M'=J, 


32  KAEL  FEIEDRICH  GAUSS 

and  also 

Then  the  equation  just  found  can  be  thrown  into  the  following  form : 

VE  .dp  .  sin  {M'—<o  +  ^)  +  VG.dq  .  sin  {M'+  »//) 

=  i/U'.  dp',  sin  M+VG'.  dq'.  sin  {M'+  w') 
or 

VE  .dp.  sin  (iV'—o)— w'+  ^\,)-VV  G  .dq  .sm  {N'—oi'+  xf,) 
=  VE' .  dp',  sin  {JST'—w')  +  VG'.  dq'.  sin  N' 

And  since  the  equation  evidently  must  be  independent  of  the  initial  direction,  this 
direction  can  be  chosen  arbitrarily.  Then,  setting  in  the  second  formula  iV'=0,  or  in 
the  first  M'=0,  we  obtain  the  following  equations  : 

VE'.  sin  0)'.  dp'=VE .  sin  (w  +  &»' — \p)  .  dp  +  V G  .  s'm  {at' — xp)  .  dq 
VG'.  sin  0}'.  dq'=VE .  sin  (i|»  —  <w)  .  dp  +  V  G  .sinxfi .  dq 

and  these  equations,  since  they  must  be  identical  with 

dp'  =  adp  +  ^  dq 

dq'=y  dp  +  S  dq 
determine  the  coefficients  a,  /8,  y,  S.     We  shall  have 
7)  l"         _    IE    sin  (w  +  w'  —  yji)    rp  _    I  G 

rf  ^'"--se' — ^^' — i^-^E^ 

-^1  (^  _  sinlt^     >^3        1^ 

T^  ^G'         sin  6)'      '      "  f  \  G' 

These  four  equations,  taken  in  connection  with  the  equations 

F  ,         F' 


cos  <u  = 


Veg'  ^"^"'-TJ^' 

EG—F^       .      ,       \E'G'—F' 


\FG—F'        .       ,        \ 


E'G' 


may  be  written 


aV{E'G'—F'^)=VEG'.sm{co  +  (o'  —  xp) 
/3V{E'G'— F")  =  VGG'.  sin  {w'  —  x},) 
yV{E'G'—F'^)^VEE'.sin{xl,—  co) 
SV{E'G'—F")  =  VGE'.sinxl, 

Since  by  the  substitutions 

dp'  =  a  dp  +  /B  dq, 
dq'^^^y  dp  -\-  hdq 


GENERAL  INYESTIGATIOITS  OF  CURVED  SURFACES  33 

the  trinomial 

E'  dp'^  +  2  F'  dp' .  dq'  +  G'  dq'^ 
is  transformed  into 

Edp''  +  2Fdp.dq  +  Gdq\ 
we  easily  obtain 

E  G  - F^  =  {E'  a'  - F'^){ah  - ^yf 

and  since,  vice  versa,  the  latter  trinomial  must  be  transformed  into  the  former  by  the 
substitution 

{a8-^y)dp^Sdp'-l3dq',     {a8-  I3y)dq  =  -ydp'  +  adq', 
we  find 

ES'-2Fyd  +  G'f=-_^,^,_jp„  -E' 

-EfiS+F{aS+/3y)-Goiy  =  ^^,Z'^],-F' 

EG  —  F^ 

E^-2Fafi-\-Go?=  ■^,^,_.^„-G' 

22. 

From  the  general  discussion  of  the  preceding  article  we  proceed  to  the  very 
extended  application  in  which,  while  keeping  for  p,  q  their  most  general  meaning,  we 
take  for  p',  q'  the  quantities  denoted  in  Ai't.  15  by  r,  cf).  We  shall  use  r,  ^  here 
also  in  such  a  way  that,  for  any  point  whatever  on  the  surface,  r  will  be  the  shortest 
distance  from  a  fixed  point,  and  <f>  the  angle  at  this  point  between  the  first  element 
of  r  and  a  fixed  direction.     We  have  thus 

E'=l,    F'=^0,    a>'=90°. 
Let  us  set  also 

i/G'==m, 

so  that  any  linear  element  whatever  becomes  equal  to 

■\/{dr^  +  m^d<f)^). 

Consequently,  the  four  equations  deduced  in  the  preceding  article  for  a,  j3,  y,  8  give 

^/E.GOs{oi  —  y|,)  =  :^ (1) 

VG.cosxl,  =  ^f (2) 


34  KARL  FRIEDRICH  GAUSS 

i/^.sin(i/»_a))=m.?i (3) 

-[/  G  ■  sill  ^  =  7n  . -^ (4) 

But  the  last  and  the  next  to  the  last  equations  of  the  preceding  article  give 
^.-^-^(|j)-2^.|.|+.@"    .      .       (5) 

{e.^-F.—).^=If.  —  -G.—).^-^         .        (6) 
'        dq  dp'     dq       \       dq  dpf     cp 

From  these  equations  must  be  determined  the  quantities  r,  (f>,  xp  and  (if  need  be) 
m,  as  functions  of  p  and  q.  Indeed,  integration  of  equation  (5)  will  give  r ;  r  being 
found,  integration  of  equation  (6)  will  give  (f) ;  and  one  or  other  of  equations  (1),  (2) 
will  give  i|»  itself.     Finally,  m  is  obtained  from  one  or  other  of  equations  (3),  (4). 

The  general  integration  of  equations  (5),  (6)  must  necessarily  introduce  two  arbi- 
trary functions.  We  shall  easily  understand  what  their  meaning  is,  if  we  remem- 
ber that  these  equations  are  not  limited  to  the  case  we  are  here  considering,  but  are 
equally  valid  if  r  and  <j)  are  taken  in  the  more  general  sense  of  Art.  16,  so  that  r  is 
the  length  of  the  shortest  Line  drawn  normal  to  a  fixed  but  arbitrary  line,  and  <f)  is 
an  arbitrary  function  of  the  length  of  that  part  of  the  fixed  line  which  is  intercepted 
between  any  shortest  line  and  an  arbitrary  fixed  point.  The  general  solution  must 
embrace  all  this  in  a  general  way,  and  the  arbitrary  functions  must  go  OA^er  into 
definite  functions  only  when  the  arbitrary  line  and  the  arbitrary  functions  of  its 
parts,  which  ^  must  represent,  are  themselves  defined.  In  our  case  an  infinitely 
small  circle  may  be  taken,  having  its  centre  at  the  point  from  which  the  distances  r 
are  measured,  and  ^  will  denote  the  parts  themselves  of  this  circle,  divided  by  the 
radius.  Whence  it  is  easily  seen  that  the  equations  (5),  (6)  are  quite  sufficient  for 
,  our  case,  provided  that  the  functions  which  they  leave  undefined  satisfy  the  condi- 
'  tion  which  r  and  <^  satisfy  for  the  initial  point  and  for  points  at  an  infinitely  small 
;  distance  from  this  point. 

Moreover,  in  regard  to  the  integration  itself  of  the  equations  (5),  (6),  we  know 
that  it  can  be  reduced  to  the  integration  of  ordinary  differential  equations,  which,  how- 
ever, often  happen  to  be  so  complicated  that  there  is  little  to  be  gained  by  the  reduc- 
tion. On  the  contrary,  the  development  in  series,  Avhich  are  abundantly  sufficient  for 
practical  requirements,  when  only  a  finite  portion  of  the  surface  is  under  considera- 
tion, presents  no   difficulty ;   and  the  formulae  thus   derived   open  a  fruitful  source  for 


GEN'EEAL  INTESTIGATIONS  OF  CUEVED  SURFACES  35 

the  solution  of  many  important  problems.  But  here  we  shall  develop  only  a  single 
example  in  order  to  show  the  nature  of  the  method. 

23. 

We  shall  now   consider  the  case  where  all  the  lines  for  which  p  is   constant  are 
shortest  lines  cutting  orthogonally  the  line  for  which  ^  =  0,  which  line  we  can  regard  (T^ 

as  the  axis  of  abscissas.  Let  A  be  the  point  for  which  r  =  0,  D  any  point  whatever  i-^^  V 
on  the  axis  of  abscissas,  AD  =  p,  B  any  point  whatever  on  the  shortest  Line  normal 
to  AD  at  D,  and  B D  =q,  so  that  p  can  be  regarded  as  the  abscissa,  q  the  ordinate 
of  the  point  B.  The  abscissas  we  assume  positive  on  the  branch  of  the  axis  of 
abscissas  to  which  ^  =  0  corresponds,  while  we  always  regard  r  as  positive.  We  take 
the  ordinates  positive  in  the  region  in  which  ^  is  measured  between  0  and  180°. 

By  the  theorem  of  Art.  16  we  shall  have 

0,-90°,    ^=0,    G  =  l, 
and  we  shall  set  also  ■ 

VE=n. 

Thus  n  wiU  be  a  function  of  p,  q,  such  that  for  q  =  Q  it  must  become  equal  to  unity. 
The  application  of  the  formula  of  Art.  18  to  our  case  shows  that  on  any  shortest 
line  whatever  we  must  have 

dd=^  ■  dp, 

dq 

where  Q  denotes  the  angle  between  the  element  of  this  line  and  the  element  of  the 
line  for  which  q  is  constant.  Now  since  the  axis  of  abscissas  is  itself  a  shortest  line, 
and  since,  for  it,  we  have  everywhere  ^  =  0,  we  see  that  for  ^  =  0  we  must  have 
everywhere 

^  =  0. 
dq 

Therefore  we  conclude  that,  if  n  is  developed  into  a  series  in  ascending  powers  of  q, 
this  series  must  have  the  following  form : 

n  =  l  -\-  f<f  -\-  gf  ■\-h<f-\-  etc. 

where  /,  g,  h,  etc.,  will  be  functions  of  p,  and  we  set 

/=/° +/>+/"/+ etc. 

A  =  A°  +  A>  +  r/  +  etc. 


/n 


36  KARL  lEIEDRICH  GAUSS 

or 

w  =  1 +/°  ?'+/>?'+/"/?'  + etc. 
-\-  g°  q^  +  g'p  q^  +  etc. 

+  A°§'*+  etc.  etc. 

24. 

The  equations  of  Art.  22  give,  in  our  case, 

.     ,       dr  dr  d^  3(k 

w  sin »/»  =  •:;— )     cos»i  =  ;r— 5      — ^^cos  li  =  m  .  :r-5     sin\h  —  m.—i-, 
^       dp  ^       dq  ^  np  ^  2q 

^dqf         \dp'  dq     dq        dp      dp 

By  the  aid  of  these  equations,  the  fifth  and  sixth  of  which  are  contained  in  the  others, 
series  can  be  developed  for  r,  <j),  \p,  ;«,  or  for  any  functions  whatever  of  these  quan- 
tities. We  are  going  to  establish  here  those  series  that  are  especially  worthy  of 
attention. 

Since  for  infinitely  small  values  of  p,  q  we  must  have 

the  series  for  r^  will  begin  with  the  terms  f'  +  (f.  We  obtain  the  terms  of  higher 
order  by  the  method  of  undetermined  coeflficients,*  by  means  of  the  equation 

\n      dp   '        \  dq  ' 
Thus  we  have 

+  q^  ^y°p'q'+ig'p'q' 

Then  we  have,  from  the  formula 

1     d{r') 


r  sin  xj)  - 


In     dp 


^^,     [2]  r^m^=p-\fyq^-\f'p'q'^-{\f"  +  i,r')p'q'     etc. 

*  We  have  thought  it  useless  to  give  the  calculation  here,  which  can  be  somewhat  abridged  by 
certain  artifices. 


GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES 
and  from  the  formula 


37 


r  Gosxp- 


2    dq 


[3] 


rcos,/,-^  +  f/>V  +  i/'/?  +(l/"-A/°')/^     etc. 

+  ig°p^(t  +  ^g'  ft 

These  formulae  give  the  angle  t/).  In  like  manner,  for  the  calculation  of  the  angle  ^, 
series  for  r  cos  <^  and  r  sin  <^  are  very  elegantly  developed  by  means  of  the  partial 
differential  equations 

9 .  r  cos  i>  I      ■     I  ■     I    d4> 

— i-  =  n  cos  <p  .  sm  i/*  —  ?■  sin  <p  .  —L 

dp  dp 

d  .  r  cos  <i  ,  ,  ■     t    dd) 

i-  =  cos  ffl  .  COS  t/;  —  r  sm  9  .  _x 

dq  dq 

9  .  r  sin  d)  •     /      •     1  1    dch  '  '^.  \ 

— i-  =  «.  sm  (/> .  sin  i//  -f-  r  COS  <^  .  -i  •   '    ' 

dp  dp 

9  .r  sin  (j>        ■     ,  ,    ,  I    d(h 

i-  =  sm  ^  .  COS  t//  +r  cos  (p  .  _ r 

95-  dq 

n  cos  t/> .  -^  4-  sin  i// .  — ?  =  0 
9  2'  9j3 

A  combination  of  these  equations  gives 

r  sin  li    9  .  r  cos  <i  ,     d  .r  cos  <i  ■  ^         ^ 

— — J- 21  ^_  r  cos  ti J-  =  r  cos  rf) 

n  dp  d  q 

r  sin  ti    d  .r  sin  d>    ,  ,     9  .  r  sin  (h  ■     , 

i 21  4-  r  cos  \b 21  =^  r  sm  (b 

n  dp  d  q 

From  these  two  equations  series  for  r  cos  <f>,  r  sin  <^  are  easily  developed,  whose  first 
terms  must  evidently  be  p,  q  respectively.      The  series  are 


%pj^Y'^^ 


■f(z)r. 


[4] 
[5] 


rcos<l.^p  +  irpq'+  ^f'p'q'  +  i^f"  -  ^/°^)  p'q'     etc. 

'ri.ih°-i^r')pq' 
rsmcf>  =  q-irp'q-if'p'q    -  {^f -^fo^)/q     etc. 


O 


-ig°p'g^-To9'ff 


From  a  combination  of  equations    [2] ,   [3] ,    [4] ,    [5]   a  series  for  r^  cos  {\p  +  <j)),  may 
be   derived,  and  from  this,  dividing  by  the  series    [1] ,  a  series  for  cos  {\jt  +  (f>),  from 


.s-f 


^tol 


V^V*"^' 


38  KARL  FRIEDRICH  GAUSS 

which  may  be  found  a  series  for  the  angle  rjt  +  (j>  itself.  However,  the  same  series 
can  be  obtained  more  elegantly  in  the  following  manner.  By  differentiating  the  first 
and  second  of  the  equations  introduced  at  the  beginning  of  this  article,  we  obtain 

,    dn  r^il;        •(3<Aa 

sm  ti \-n  cos  li  .  — -f  -4-  sm  li) .  — i:  =  U 

dq  dq  dp 


and  this  combined  with  the  equation 


n  cos  t|» .  — ?  +  sin  i/> .  -^  =  0 
dq  ^         ^     32? 


\1- 


gives 


V^  n         dq  n  dp  dq 

From  this  equation,  by  aid  of  the  method  of  undetermined  coefficients,  we  can  easily 
derive  the  series  for  i//  +  </>,  if  we  observe  that  its  first  term  must  be  ^  it,  the  radius 
being  taken  equal  to  unity  and  2  tt  denoting  the  circumference  of  the  circle, 

[6]  4>  +  <}>  =  ^^-rF9-U'f9~{U"-ifn/9     etc. 

-ff°pq'-i(/'p'q'' 

It  seems  worth  while  also  to  develop  the  area  of  the  triangle  A  B  D  into  a  series. 
For  this  development  we  may  use  the  following  conditional  equation,  which  is  easily 
derived  from  sufficiently  obvious  geometric  considerations,  and  in  which  S  denotes  the 
required  area : 

!  '^  r  s,va.yh   dS  dS      rsm\b     f     ■, 

C  ,A •  ■■">  ^ •  -^  +  r  cos  t/; .  r-  = ^  -indq 

\ij^   \  n         dp  ^    dq  n        ^ 

the  integration  beginning  with  q  —  0.  From  this  equation  we  obtain,  by  the  method 
of  undetermined  coefficients, 

[7]  S'=Xpg-A^fOp^q-^fp*q-(^f"-^fo^)p^g       ,t,_ 


GENERAL  E^VESTIGATIONS  OP  CURVED  SURFACES  39 

25. 

From  the  formulse  of  the  preceding  article,  which  refer  to  a  right  triangle  formed 
by  shortest  lines,  we  proceed  to  the  general  case.  Let  C  be  another  point  on  the 
same  shortest  line  D  B,  for  which  point  p  remains  the  same  as  for  the  point  B,  and 
q',  r',  (f)',  \\i',  S'  have  the  same  meanings  as  q,  r,  (jj,  i//,  S  have  for  the  point  B.  There 
will  thus  be  a  triangle  between  the  points  A,  B,  C,  whose  angles  we  denote  by 
A,  B,  C,  the  sides  opposite  these  angles  by  a,  h,  c,  and  the  area  by  cr.  We  represent 
the  measure  of  curvature  at  the  points  A,  B,  C  by  a,  fi,  y  respectively.  And  then 
supposing  (which  is  permissible)  that  the  quantities  p,  q,  q  —  q'  are  positive,  we  shall 
have 

A  =  (f>-<p',     B^xp,     C=7r  — f, 

a  =  q  —  q',      b  —  r',      c  =  r,         a-=S—S'. 

We  shall  first  express  the  area  <t  by  a  series.  By  changing  in  [7]  each  of  the 
quantities  that  refer  to  B  into  those  that  refer  to  C,  we  obtain  a  formula  for  S'. 
Whence  we  have,  exact  to  quantities  of  the  sixth  order, 

<r  =  ip{q-q')il-ir{p'+q'+qq'  +  q'') 

--hf'P  (6/+  7  q^+  Iqq'-^l  q'^) 

This  formula,  by  aid  of  series  [2] ,  namely,  ^      I  (       (^  " j 

csmB^p{l-if°q'-^fpq'-iff°q'-etG.)  '  ' 

can  be  changed  into  the  following : 

(r  =  ^acsmB(l—lf°{p''—q^+qq'+q'^) 

-iof'piQf-8q'+7qq'+7q'') 

-^9°  {^P^q  +  ip^q'-  Qp^+Aq'q'+A  qq"+  4  q")) 

The  measure  of  curvature  for  any  point  whatever  of  the  surface  becomes  (by  Art. 
19,  where  m,  p,  q  were  what  n,  q,  p  are  here) 

k---    g'^  _      2/+6^^  +  12^g^  +  etc. 
n    dq^  ^  1  +/§^  +  etc. 

-=  -  2/-  Qffq-{12h~  2f)  q"-  -  etc. 

Therefore  we  have,  when  p,  q  refer  to  the  point  5, 

/8  =  -  2/°  -  2/>  ~^g°q-  If'p''  -^g'pq-  (12  /j°  -  2/°^)  q"-  -  etc. 


-V 
> 


40  KARL  FRIEDRICH  GAUSS 

Also 

y  =  -2f°-  2f'p  -  6  g°q'  -  2f"f  -  6  g'pq'  -  (12  h°  -  2f°^)  q'^  -  etc. 
a  =  -2/° 

Introducing  these  measures  of  curvature  into  the  expression  for  cr,  we  obtain  the  fol- 
lowing expression,  exact  to  quantities  of  the  sixth  order  (exclusive) : 

cr  =  ^ac  sinB  (I  +  -^a{4:p'  -2q'  +  S  qq'  +  S  q") 
+  -^^{Sp^-Qq'+Qqq'  +  Sq") 
+  Tior(3/-2^^+      qq'  +  iq'^)) 

The  same  precision  will  remain,  if  for  p,  q,  q'  we  substitute  c  sin  B,  e  cos  B,  c  cos  B  —  a. 
This  gives 

^         [8]  o-  =  ^ftcsin  jB  (1  +  y^a(3a^  + 4c^—    9  accost) 

hr  \  +  ylo  y8  (3  a'  +  3  c''  —  12  ac  cos  B) 

^^t    W^  _1- _i_^  y  (4  ^2  +  3  ^2  _    9«ccos^)) 

Since  all  expressions  which  refer  to  the  line  AD  drawn  normal  to  B  C  have  disap- 
peared from  this  equation,  we  may  permute  among  themselves  the  points  A,  B,  C  and 
the  expressions  that  refer  to  them.     Therefore  we  shall  have,  with  the  same  precision, 

[9]  a-  =  ^bc  sin  A  (1  +  T^  a  {S  P  +  3  c^  -12  be  cos  A) 

+  _i_y3(3J2  +  4p2_    QiccosA) 

+  ^y(4J2  +  3c'-    9^ccos^)) 
[10]  o-  =  ia5sinC(l  +  y^a(3a^  +  4J^-    9  ab  cos  C) 

+  ^^(4^2  + 3  5^-    9  ab  cos  C) 
+  ^y{Sa^  +  Sb^  — 12  ab  cos  C)) 

26. 

The  consideration  of  the  rectilinear  triangle  whose  sides  are  equal  to  a,  b,  c  is  of 
great  advantage.  The  angles  of  this  triangle,  which  we  shall  denote  by  A*,  B*,  C*, 
differ  from  the  angles  of  the  triangle  on  the  curved  surface,  namely,  from  A,  B,  C, 
by  quantities  of  the  second  order ;  and  it  will  be  worth  while  to  develop  these  differ- 
ences accurately.  However,  it  will  be  sufficient  to  show  the  first  steps  in  these  more 
tedious  than  difficult  calculations. 

Replacing  in  formulae  [1],  [4],  [5]  the  quantities  that  refer  to  B  by  those  that 
refer  to  C,  we  get  formuke  for  r'^,  r'cos<f>',  r' sin  <^'.  Then  the  development  of  the 
expression 


GENEEAL  ENTESTIGATIOlSrS  OF  CrRVED  SUEFACES  41 

yS.  -j_  ^/2_  ^^  _  ^/j2  _  2  ^  COS  (j) .  r'  cos  (f)'  —  2  r  sin  <^  .  r'  sin  <j>' 

=  b^  +  c''- a'' -2  be  cos  A  '  -"^ '  t*1  ^  T 

=  2  5  c  (cos  A*  —  cos  J.),  ;,  ' 

combined  with  the  development  of  the  expression 

r  sin <f> .  r'  cos  <f>'  —  r  cos  <j) .  r'  sin  (f)'  =  bc  sin  A, 
gives  the  following  formula : 

cos  A*  —  COS  A  ~  —  {q  —  q')p  sin  J.  (  ^/°  +  -!•/'  jo  +  \g°  {q  +  §'') 

+  (  i  ^°- 9^/°^)  (?^  +  ^?'+ ?")  +  etc.) 
From  this  we  have,  to  quantities  of  the  fifth  order, 

A*-A^  +  {q- q')p  (i/°  +  i/>  +  i^°  (?  +  q')  +  iV/"/ 

Combining  this  formula  with 

2(T  =  ap(l-  \f°  {f-  ^q'  +  qq'  +  q'^)-  etc.) 

and  with  the  values  of  the  quantities  a,  /8,  y  found  in  the  preceding  article,  we  obtain, 

to  quantities  of  the  fifth  order,  n  j 

[11]  A*=A-ai^a  +  i^^  +  ^'^  +  -^f"p'  +  \g'p{q  +  q')  ^^^  ^  67'"   ' 

+  \h°{?>q^-2qq'+?,q'^)  ' 

By  precisely  similar  operations  we  derive  „.    ^  .  "1  ^ 

[12]  B^^B-<Ti-^a  +  \fi-^i^y  +  i^f"p'^+i^g'p{2q^-q')  S^-^    f ' 


+  \h°  {iq^-iqq'  +  ?>q' 


-9VrM2/+8?^-8^/  +  ll^-))  O^S^'^'' 

[13]  ^*  =  C-cr(TLa  +  TV/3  +  ir  +  Tior/+iV/i'(^  +  2?')  1^  \' 

+  \h°{2,q'-iqq'  +  'kq'^) 
--hr\2p^  +  \\q--%qq'  +  %q'^)) 

From  these  formulae   we   deduce,  since  the   sum  A*  +  B*  +  C*   is   equal  to  two  right 
angles,  the  excess  of  the  sum  A-\-B-\-C  over  two  right  angles,  namely, 

[14]  A+B  +  C=^+<Ti\a  +  \fi  +  \y  +  \f"p'+^g'p{q  +  q')         "^     |  '   ^^  ' 

+  {2h°-y°'){q'-qq'+q'')) 

This  last  equation  could  also  have  been  derived  from  formula  [6]. 


42  KAEL  FRIEDRICH  GAUSS 

27. 

^.       n  If  the  curved  surface  is  a  sphere  of  radius  R,  we  shall  have 

.        l^        ^'  1 


or 


1 


24  i?* 
Consequently,  formula  [14]  becomes 

which  is  absolutely  exact.     But  formulae  [11],  [12],  [13]  give 

or,  with  equal  exactness, 

^*  ""  ^  ~  O"^  ~  T80^(*'  +  *'  -  2  (^) 

Neglecting  quantities   of  the  fourth  order,  we  obtain  from  the  above   the  well-known 
theorem  first  established  by  the  illustrious  Legendre. 

28. 

Our  general  formuli3e,  if  we  neglect  terms  of  the  fourth  order,  become  extremely 
simple,  namely  : 


GENERAL  INVESTIGATIONS  OF  CUEVED  SURFACES  43 

Thus  to  the  angles  A,  B,  C  on  a  non-spherical  surface,  unequal  reductions  must 
be  applied,  so  that  the  sines  of  the  changed  angles  become  proportional  to  the  sides 
opposite.  The  inequality,  generally  speaking,  will  be  of  the  third  order;  but  if  the 
surface  differs  Little  from  a  sphere,  the  inequality  will  be  of  a  higher  order.  Even  in 
the  greatest  triangles  on  the  earth's  surface,  whose  angles  it  is  possible  to  measure, 
the  difference  can  always  be  regarded  as  insensible.  Thus,  e.  g.,  in  the  greatest  of 
the  triangles  which  we  haA^e  measured  in  recent  years,  namely,  that  between  the 
points  Hohehagen,  Brocken,  Inselberg,  where  the  excess  of  the  sum  of  the  angles  was 
14". 85348,  the  calculation  gave  the  following  reductions  to  be  applied  to  the  angles  : 

Hohehagen -4".95113 

Brocken — 4".95104 

Inselberg — 4".95131. 

29. 

We  shall  conclude  this  study  by  comparing  the  area  of  a  triangle  on  a  curved 
surface  with  the  area  of  the  rectilinear  triangle  whose  sides  are  a,  h,  c.  We  shall 
denote  the  area  of  the  latter  by  cr* ;    hence 

cr*  =  ^5c  sin  J[*  =  ^ac  sin^*  —  ^ab  sin  0* 

We  have,  to  quantities  of  the  fourth  order, 

sin  A*=sm  A  —  ^^  a  cos  A  .  {2  a  +  fi  +  y) 

or,  with  equal  exactness, 

sinJ.  =  sin^*.(l  +  -^bc  cos  A  .  {2  a  + 13  + y)) 

Substituting  this  value  in  formula  [9],  we  shall  have,  to  quantities  of  the  sixth  order, 

a-  =  i  be  sin  A"" .  (1  +  ^  a  {8  b''+  5  c^-2bc  cos  A) 
+  -ih/^  (3  5^+  4  c^-  4  Jc  cos  A) 
+  j^y  {A  b'+  5  c^-  4.bc  cos  A)}, 

or,  with  equal  exactness, 

o-  =  cr*  (1+T^a  {a'+  2  b'+  2  c')  +rhl^i^  «'+  ^'+  2  c^)  +  yi^y  (2  a'+  2  b'+  e')) 
For  the  sphere  this  formula  goes  over  into  the  following  form : 

o-  =  cr*  (1  +  2V  a  («'+  i"'+  c'))- 


5i^ap>7? 


44  KARL  FRIEDRICH  GAUSS 

It  is  easily  verified  that,  with  the  same  precision,  the  following  formula  may  be  taken 
instead  of  the  above  : 

I  sin  ^    .  sin  B    .  sin  C 


'^ 


sin  A*  .  sin  B* .  sin  C* 


If  this  formula  is  applied  to  triangles  on  non-spherical  curved  surfaces,  the  error,  gen- 
erally speaking,  will  be  of  the  fifth  order,  but  will  be  insensible  in  all  triangles  such 
as  may  be  measured  on  the  earth's  surface. 


GAUSS'S  AESTRACT  45 


GAUSS'S   ABSTRACT   OF   THE   DISQUISITIONES   GENERALES   CIRCA 

SUPERFICIES   CURVAS,   PRESENTED   TO   THE   ROYAL 

SOCIETY   OF   GOTTINGEN. 


GoTTiNGiscHE  GELEHRTE  Anzeigen.    No.  177.     Pages  1761-1768.     1827.     November  5. 


On  the  8th  of  October,  Hofrath  Gauss  presented  to  the  Royal  Society  a  paper : 
Disquisitiones  generales  circa  superficies  curvas. 

Although  geometers  have  given  much  attention  to  general  investigations  of  curved 
surfaces  and  their  results  cover  a  significant  portion  of  the  domain  of  higher  geometry, 
this  subject  is  still  so  far  from  being  exhausted,  that  it  can  well  be  said  that,  up  to 
this  time,  but  a  small  portion  of  an  exceedingly  fruitful  field  has  been  cultivated. 
Through  the  solution  of  the  problem,  to  find  all  representations  of  a  given  surface  upon 
another  in  which  the  smallest  elements  remain  unchanged,  the  author  sought  some 
years  ago  to  give  a  new  phase  to  this  study.  The  purpose  of  the  present  discussion 
is  further  to  open  up  other  new  points  of  view  and  to  develop  some  of  the  new  truths 
which  thus  become  accessible.  We  shall  here  give  an  account  of  those  things  which 
can  be  made  intelligible  in  a  few  words.  But  we  wish  to  remark  at  the  outset  that 
the  new  theorems  as  weU  as  the  presentations  of  new  ideas,  if  the  greatest  generality 
is  to  be  attained,  are  still  partly  in  need  of  some  limitations  or  closer  determinations, 
which  must  be  omitted  here. 

In  researches  in  which  an  infinity  of  directions  of  straight  lines  in  space  is  con- 
cerned, it  is  advantageous  to  represent  these  directions  by  means  of  those  points  upon 
a  fixed  sphere,  which  are  the  end  points  of  the  radii  drawn  parallel  to  the  lines.  The 
centre  and  the  radius  of  this  auxiliary  sphere  are  here  quite  arbitrary.  The  radius  may 
be  taken  equal  to  unity.  This  procedure  agrees  fundamentally  with  that  which  is  con- 
stantly employed  in  astronomy,  where  all  directions  are  referred  to  a  fictitious  celestial 
sphere  of  infinite  radius.  Spherical  trigonometry  and  certain  other  theorems,  to  which 
the  author  has  added  a  new  one  of  frequent  application,  then  serve  for  the  solution  of 
the  problems  which  the  comparison  of  the  various  directions  involved  can  present. 


46  GAUSS'S  ABSTRACT 

If  Tve  represent  the  direction  of  the  normal  at  each  point  of  the  curved  surtace  by 
the  corresponding  point  of  the  sphere,  determined  as  above  indicated,  namely,  in  this 
way,  to  every  point  on  the  surface,  let  a  point  on  the  sphere  correspond;  then,  gener- 
ally speaking,  to  eA^ery  line  on  the  curved  surface  will  correspond  a  line  on  the  sphere, 
and  to  every  part  of  the  former  surface  will  correspond  a  part  of  the  latter.  The  less 
this  part  differs  from  a  plane,  the  smaller  will  be  the  corresponding  part  on  the  sphere. 
It  is,  therefore,  a  very  natural  idea  to  use  as  the  measure  of  the  total  curvature, 
which  is  to  be  assigned  to  a  part  of  the  curved  surface,  the  area  of  the  corresponding 
part  of  the  sphere.  For  this  reason  the  author  calls  this  area  the  integral  curvature  of 
the  corresponding  part  of  the  curved  surface.  Besides  the  magnitude  of  the  part,  there 
is  also  at  the  same  time  its  position  to  be  considered.  And  this  position  may  be  in 
the  two  parts  similar  or  inverse,  quite  independently  of  the  relation  of  their  magni- 
tudes. The  two  cases  can  be  distinguished  by  the  positive  or  negative  sign  of  the 
total  curvature.  This  distinction  has,  however,  a  definite  meaning  only  when  the 
figures  are  regarded  as  upon  definite  sides  of  the  two  surfaces.  The  author  regards 
the  figure  in  the  case  of  the  sphere  on  the  outside,  and  in  the  case  of  the  curved  sur- 
face on  that  side  upon  which  we  consider  the  normals  erected.  It  follows  then  that 
the  positive  sign  is  taken  in  the  case  of  convexo-convex  or  concavo-concave  surfaces 
(which  are  not  essentially  different),  and  the  negative  in  the  case  of  concaA^o-convex 
surfaces.  If  the  part  of  the  curved  surface  in  question  consists  of  parts  of  these  differ- 
ent sorts,  still  closer  definition  is  necessary,  Avhich  must  be  omitted  here. 

The  comparison  of  the  areas  of  two  corresponding  parts  of  the  curved  surface  and  of 
the  sphere  leads  now  (in  the  same  manner  as,  e.  g.,  from  the  comparison  of  volume  and 
mass  springs  the  idea  of  density)  to  a  new  idea.  The  author  designates  as  measure  of 
curvature  at  a  point  of  the  curved  surface  the  A^alue  of  the  fraction  whose  denominator  is 
the  area  of  the  infinitely  small  part  of  the  curved  surface  at  this  point  and  whose  numer- 
ator is  the  area  of  the  corresponding  part  of  the  surface  of  the  auxiliary  sphere,  or  the 
integral  curvature  of  that  element.  It  is  clear  that,  according  to  the  idea  of  the  author, 
integral  curvature  and  measure  of  curvature  in  the  case  of  curved  surfaces  are  analo- 
gous to  what,  in  the  case  of  curved  lines,  are  called  respectively  amplitude  and  curva- 
ture simply.  He  hesitates  to  apply  to  curved  surfaces  the  latter  expressions,  which 
have  been  accepted  more  from  custom  than  on  account  of  fitness.  Moreover,  less 
depends  upon  the  choice  of  words  than  upon  this,  that  their  introduction  shall  be  justi- 
fied by  pregnant  theorems. 

The  solution  of  the  problem,  to  find  the  measure  of  curvature  at  any  point  of  a  curved 
surface,  appears  in  different  forms  according  to  the  manner  in  which  the  nature  of  the 
curved  surface  is  given.      When  the  points  in  space,  in  general,  are   distinguished  by 


GAUSS'S  ABSTRACT  47 

three  rectangular  coordinates,  the  simplest  method  is  to  express  one  coordinate  as  a  func- 
tion of  the  other  two.  In  this  way  we  obtain  the  simplest  expression  for  the  measure  of 
curvature.  But,  at  the  same  time,  there  arises  a  remarkable  relation  between  this 
measure  of  curvature  and  the  curvatures  of  the  curves  formed  by  the  intersections  of 
the  curved  surface  with  planes  normal  to  it.  Euler,  as  is  well  known,  first  showed 
that  two  of  these  cutting  planes  which  intersect  each  other  at  right  angles  have  this 
property,  that  in  one  is  found  the  greatest  and  in  the  other  the  smallest  radius  of  cur- 
vature ;  or,  more  correctly,  that  in  them  the  two  extreme  curvatures  are  found.  It  wiU 
follow  then  from  the  above  mentioned  expression  for  the  measure  of  curvature  that  this 
will  be  equal  to  a  fraction  whose  numerator  is  unity  and  whose  denominator  is  the  product 
of  the  extreme  radii  of  curvature.  The  expression  for  the  measure  of  curvature  will  be 
less  simple,  if  the  nature  of  the  curved  surface  is  determined  by  an  equation  in  x,  y,  z. 
And  it  wiU  become  still  more  complex,  if  the  nature  of  the  curved  surface  is  given  so  that 
X,  y,  z  are  expressed  in  the  form  of  functions  of  two  new  variables  p,  q.  In  this  last  case 
the  expression  involves  fifteen  elements,  namely,  the  partial  differential  coefficients  of  the 
first  and  second  orders  of  x,  y,  z  with  respect  to  p  and  q.  But  it  is  less  important  in  itself 
than  for  the  reason  that  it  facilitates  the  transition  to  another  expression,  which  must  be 
classed  with  the  most  remarkable  theorems  of  this  study.  If  the  nature  of  the  curved 
surface  be  expressed  by  this  method,  the  general  expression  for  any  linear  element  upon 
it,  or  for  Vidi?  +  i/  +  ^s'),  has  the  form  V{Edf  +  2Fdp.dq  +  a  dq\  where  E,  F,  a 
are  again  functions  of  p  and  q.  The  new  expression  for  the  measure  of  curvature  men- 
tioned above  contains  merely  these  magnitudes  and  their  partial  differential  coefficients 
of  the  first  and  second  order.  Therefore  we  notice  that,  in  order  to  determine  the 
measure  of  curvature,  it  is  necessary  to  know  only  the  general  expression  for  a  Hnear 
element ;  the  expressions  for  the  coordinates  x,  y,  z  are  not  required.  A  direct  result 
from  this  is  the  remarkable  theorem  :  If  a  curved  surface,  or  a  part  of  it,  can  be  devel- 
oped upon  another  surface,  the  measure  of  curvature  at  every  point  remains  unchanged 
after  the  development.  In  particular,  it  follows  from  this  further :  Upon  a  curved 
surface  that  can  be  developed  upon  a  plane,  the  measure  of  curvature  is  everywhere 
equal  to  zero.  From  this  we  derive  at  once  the  characteristic  equation  of  surfaces 
developable  upon  a  plane,  namely. 


dx"' dy''       \dx.dy 


\dx.dyf  ' 


when  z  is  regarded  as  a  function  of  x  and  y.  This  equation  has  been  known  for  some 
time,  but  according  to  the  author's  judgment  it  has  not  been  established  previously 
with  the  necessary  rigor. 


48  GAUSS'S  ABSTRACT 

These  theorems  lead  to  the  consideration  of  the  theory  of  curved  surfaces  from  a 
new  point  of  view,  where  a  wide  and  still  wholly  uncultivated  field  is  open  to  investi- 
gation. If  we  consider  surfaces  not  as  boundaries  of  bodies,  but  as  bodies  of  which 
one  dimension  vanishes,  and  if  at  the  same  time  we  conceive  them  as  flexible  but  not 
extensible,  we  see  that  two  essentially  different  relations  must  be  distinguished,  namely, 
on  the  one  hand,  those  that  presuppose  a  definite  form  of  the  surface  in  space ;  on  the 
other  hand,  those  that  are  independent  of  the  various  forms  which  the  surface  may 
assume.  This  discussion  is  concerned  with  the  latter.  In  accordance  with  what  has 
been  said,  the  measure  of  curvature  belongs  to  this  case.  But  it  is  easily  seen  that 
the  consideration  of  figures  constructed  upon  the  surface,  their  angles,  their  areas  and 
their  integral  curvatures,  the  joining  of  the  points  by  means  of  shortest  Hnes,  and  the 
like,  also  belong  to  this  case.  AU  such  investigations  must  start  from  this,  that  the 
very  nature  of  the  curved  surface  is  given  by  means  of  the  expression  of  any  linear 
element  in  the  form  V {E dp^-\-  2  F dp  .dq-\-  G  dq^).  The  author  has  embodied  in  the 
present  treatise  a  portion  of  his  investigations  in  this  field,  made  several  years  ago, 
while  he  limits  himself  to  such  as  are  not  too  remote  for  an  introduction,  and  may,  to 
some  extent,  be  generally  helpful  in  many  further  investigations.  In  our  abstract,  we 
must  limit  ourselves  still  more,  and  be  content  with  citing  only  a  few  of  them  as 
types.     The  following  theorems  may  serve  for  this  purpose. 

If  upon  a  curved  surface  a  system  of  infinitely  many  shortest  lines  of  equal  lengths 
be  drawn  from  one  initial  point,  then  will  the  line  going  through  the  end  points  of 
these  shortest  lines  cut  each  of  them  at  right  angles.  If  at  every  point  of  an  arbitrary 
line  on  a  curved  surface  shortest  lines  of  equal  lengths  be  drawn  at  right  angles  to  this 
line,  then  will  all  these  shortest  lines  be  perpendicular  also  to  the  line  which  joins  their 
other  end  points.  Both  these  theorems,  of  which  the  latter  can  be  regarded  as  a  gen- 
eralization of  the  former,  wiU  be  demonstrated  both  analytically  and  by  simple  geomet- 
rical considerations.  The  excess  of  the  sum  of  the  angles  of  a  triangle  formed  hy  shortest  lines 
over  two  right  angles  is  equal  to  the  total  curvature  of  the  triangle.  It  will  be  assumed  here 
that  that  angle  (57°  17' 45")  to  which  an  arc  equal  to  the  radius  of  the  sphere  corresponds 
will  be  taken  as  the  unit  for  the  angles,  and  that  for  the  unit  of  total  curvature  wiU  be 
taken  a  part  of  the  spherical  surface,  the  area  of  which  is  a  square  whose  side  is  equal  to 
the  radius  of  the  sphere.  Evidently  we  can  express  this  important  theorem  thus  also  : 
the  excess  over  two  right  angles  of  the  angles  of  a  triangle  formed  by  shortest  lines  is  to 
eight  right  angles  as  the  part  of  the  surface  of  the  auxiliary  sphere,  which  corresponds 
to  it  as  its  integral  curvature,  is  to  the  whole  surface  of  the  sphere.  In  general,  the 
excess  over  2  w  —  4  right  angles  of  the  angles  of  a  polygon  of  n  sides,  if  these  are 
shortest  lines,  will  be  equal  to  the  integral  curvature  of  the  polygon. 


GAUSS'S   ABSTEAOT  49 

The  general  investigations  developed  in  this  treatise  will,  in  the  conclusion,  be  applied 
to  the  theory  of  triangles  of  shortest  lines,  of  which  we  shall  introduce  only  a  couple  of 
important  theorems.  If  a,  b,  c  be  the  sides  of  such  a  triangle  (they  will  be  regarded  as 
magnitudes  of  the  first  order) ;  ^,  5,  C  the  angles  opposite ;  a,  j8,  y  the  measures  of 
curvature  at  the  angular  points ;  a  the  area  of  the  triangle,  then,  to  magnitudes  of  the 
fourth  order,  i  (a  + /3  +  y)  cr  is  the  excess  of  the  sum  A^-B  +  C  over  two  right  angles. 
Further,  with  the  same  degree  of  exactness,  the  angles  of  a  plane  rectilinear  triangle 
whose  sides  are  a,  h,  c,  are  respectively 


^-J5(2a+y8  +  r) 
C-TL(a+^+2y) 


We  see  immediately  that  this  last  theorem  is  a  generalization  of  the  familiar  theorem  first 
established  by  Legendre.  By  means  of  this  theorem  we  obtain  the  angles  of  a  plane 
triangle,  correct  to  magnitudes  of  the  fourth  order,  if  we  diminish  each  angle  of  the  cor- 
responding spherical  triangle  by  one-third  of  the  spherical  excess.  In  the  case  of  non- 
spherical  surfaces,  we  must  apply  unequal  reductions  to  the  angles,  and  this  inequality, 
generally  speaking,  is  a  magnitude  of  the  third  order.  However,  even  if  the  whole  sur- 
face differs  only  a  little  from  the  spherical  form,  it  will  still  involve  also  a  factor  denoting 
the  degree  of  the  deviation  from  the  spherical  form.  It  is  unquestionably  important  for 
the  higher  geodesy  that  we  be  able  to  calculate  the  inequalities  of  those  reductions  and 
thereby  obtain  the  thorough  conviction  that,  for  aU  measurable  triangles  on  the  surface 
of  the  earth,  they  are  to  be  regarded  as  quite  insensible.  So  it  is,  for  example,  in  the 
case  of  the  greatest  triangle  of  the  triangulation  carried  out  by  the  author.  The  greatest 
side  of  this  triangle  is  ahnost  fifteen  geographical*  miles,  and  the  excess  of  the  sum 
of  its  three  angles  over  two  right  angles  amounts  almost  to  fifteen  seconds.  The  three 
reductions  of  the  angles  of  the  plane  triangle  are  4".95113,  4".9ol04,  4".95131.  Besides, 
the  author  also  developed  the  missing  terms  of  the  fourth  order  in  the  above  expres- 
sions. Those  for  the  sphere  possess  a  very  simple  form.  However,  in  the  case  of 
measurable  triangles  upon  the  earth's  surface,  they  are  quite  insensible.  And  in  the 
example  here  introduced  they  would  have  diminished  the  first  reduction  by  only  two 
units  in  the  fifth  decimal  place  and  increased  the  third  by  the  same  amount. 


*  This  German  geographical  mile  is  four  minutes  of  arc  at  the  equator,  namely,  7.42  kilome- 
ters, and  is  equal  to  about  4.6  English  statute  miles.     [Translators.] 


NOTES  51 


NOTES.  -^"^J^'j^fy 

Art.  1,  p.  3,  1.  3.     Gauss  got  the  idea  of  using  the  auxiliary  sphere  from  astron-  U^\,  -^^  k^   A/' 

omy.     Cf.  Gauss's  Abstract,  p.  45.  > /y/      ^      v^   X.^ 

Art.  2,  p.   3,  1.  2  fr.  hot.      In  the  Latin  text  situs  is  used  for  the   direction  or        W"    .  ^        /^        y 
orientation  of  a  plane,  the  position  of  a  plane,  the  direction  of  a  line,  and  the  posi-        C^]}-^    ' 
tion  of  a  point.  v/jj/" 


P' 


Art.  2,  p.  4,  1.  14.      In  the  Latin  texts  the  notation  '    ,   ^^     l^y^^ 

cos  (1)  r  +  cos  (2)  L'  +  cos  (3)  L'^1  ■   r^  ^        ^»^        '^ 

is  used.     This  is  replaced  in  the  translations   (except   Boklen's)   by  the  more   recent  ^  -^ .  Vk/^"'  ^ 

cos^(l)X  +  cos^(2)i;  +  cosn3)X  =  l.  -  ^    fj^    '^^ 

Art.  2,  p.  4,  1.  3  fr.  bot.      This  stands  in  the  original  and  in  Liouville  s  reprint,     '  >  ,jj<^^                  ''"^   '■ 

cos  A  (cos  i  sin  t'  —  sin  t  cos  f )  (cos  t"  sin  f "  —  sin  t"  sin  t'").  ^  ^ ,  V-«>^     ' 

Art.  2,  pp.  4-6.      Theorem  VI  is   original  with   Gauss,  as  is  also  the  method  of  <      0  .■   i"^  '^'  ' 

deriving  VII.      The  following  figures  show  the  points  and  lines   of  Theorems  VI  and  "^    ^  .  v-"*- 

VII:  (^"^ 


Art.  3,  p.  6.  The  geometric  condition  here  stated,  that  the  curvature  be  continu- 
ous for  each  point  of  the  surface,  or  part  of  the  surface,  considered  is  equivalent  to 
the  analytic  condition  that  the  first  and  second  derivatives  of  the  function  or  func- 
tions defining  the  surface  be  finite  and  continuous  for  aU  points  of  the  surface,  or 
part  of  the  surface,  considered. 

Art.  4,  p.  7,  1.  20.     In  the  Latin  texts  the  notation  XX  for  X"^,  etc.,  is  used. 


52  NOTES 

Art.  4,  p.  7.  "  The  second  method  of  representing  a  surface  (the  expression  of 
the  coordinates  by  means  of  two  auxiliary  variables)  was  first  used  by  Gauss  for 
arbitrary  surfaces  in  the  ease  of  the  problem  of  conformal  mapping.  [Astronomische 
Abhandlungen,  edited  by  H.  C.  Schumacher,  vol.  Ill,  Altona,  1825 ;  Gauss,  Werke, 
vol.  IV,  p.  189  ;  reprinted  in  vol.  55  of  Ostwald's  Klassiker. — Cf.  also  Gauss,  Theoria 
attractionis  corporum  sphaer.  ellipt.,  Comment.  Gott.  II,  1813  ;  Gauss,  Werke,  vol.  V, 
p.  10.]  Here  he  applies  this  representation  for  the  first  time  to  the  determination  of 
the  direction  of  the  surface  normal,  and  later  also  to  the  study  of  curvature  and  of 
geodetic  lines.  The  geometrical  significance  of  the  variables  p,  q  is  discussed  more  fully 
in  Art.  17.  This  method  of  representation  forms  the  source  of  many  new  theorems, 
of  which  these  are  particularly  worthy  of  mention :  the  corollary,  that  the  measure  of 
curvature  remains  unchanged  by  the  bending  of  the  surface  (Art.  11,  12) ;  the  theorems 
of  Art.  15,  16  concerning  geodetic  lines ;  the  theorem  of  Art.  20 ;  and,  finally,  the 
results  derived  in  the  conclusion,  which  refer  a  geodetic  triangle  to  the  rectilinear  trian- 
gle whose  sides  are  of  the  same  length."      [Wangerin.] 

Art.  5,  p.  8.  "To  decide  the  question,  which  of  the  two  systems  of  values  found 
in  Art.  4  for  X,  Y,  Z  belong  to  the  normal  directed  outwards,  which  to  the  normal 
directed  inwards,  we  need  only  to  apply  the  theorem  of  Art.  2  (VII),  provided  we  use 
the  second  method  of  reiiresenting  the  surface.  If,  on  the  contrary,  the  surface  is 
defined  by  the  equation  between  the  coordinates  TF=  0,  then  the  following  simpler  con- 
considerations  lead  to  the  answer.  We  draw  the  line  da-  from  the  point  A  towards 
the  outer  side,  then,  if  dx,  dy,  dz  are  the  projections  of  d<j,  we  have 

Pdx^  Qdy  -^  Rdz>Q. 

On  the  other  hand,  if  the  angle  between  o-  and  the  normal  taken  outward  is  acute, 
then 

^X  +  ^Y+^Z>^. 
da-  acT  acr 

This  condition,  since  da-  h  positive,  must  be  combined  with  the  preceding,  if  the  first 
solution  is  taken  for  X,  Y,  Z.  This  result  is  obtained  in  a  similar  way,  if  the  sur- 
face is  analytically  defined  by  the  third  method."      [Wangerin.] 

Art.  6,  p.  10,  1.  4.  The  definition  of  measure  of  curvature  here  given  is  the  one 
generally  used.  But  Sophie  Germain  defined  as  a  measure  of  curvature  at  a  point  of 
a  surface  the  sum  of  the  reciprocals  of  the  principal  radii  of  curvature  at  that  point, 
or  double  the  so-called  mean  curvature.  Cf.  Crelle's  Journ.  fiir  Math.,  vol.  VII. 
Casorati  defined  as  a  measure  of  curvature  one-half  the  sum  of  the  squares  of  the 
reciprocals  of  the  principal  radii  of  curvature  at  a  point  of  the  surface.  Cf.  Rend. 
del  R.  Istituto  Lombardo,  ser.  2,  vol.  22,  1889  ;  Acta  Mathem.  vol.  XIV,  p.  95,  1890. 


NOTES  53 

Art.  6,  p.  11,  1.  21.  Gauss  did  not  carry  out  his  intention  of  studying  the  most 
general  cases  of  figures  mapped  on  the  sphere. 

Art.  7,  p.  11,  1.  31.  "  That  the  consideration  of  a  surface  element  which  has  the 
form  of  a  triangle  can  be  used  in  the  calculation  of  the  measure  of  curvature,  follows 
from  this  fact  that,  according  to  the  formula  developed  on  page  12,  k  is  independent 
of  the  magnitudes  dx,  dy,  S.r,  hy,  and  that,  consequently,  h  has  the  same  value  for 
every  infinitely  small  triangle  at  the  same  point  of  the  surface,  therefore  also  for  sur- 
face elements  of  any  form  whatever  lying  at  that  point."      [Wangerin.] 

Art.  7,  p.  12,  1.  20.     The  notation  in  the  Latin  text  for  the  partial  derivatives : 

dX     dX 

~j — '      'j — '     6tc., 

ax       ay  ' 

has  been  replaced  throughout  by  the  more  recent  notation : 

dX    dX 

^5 — 5    "5 — J    etc. 
dz       cy 

Art.  7,  p.  13,  1.  16.  This  formula,  as  it  stands  in  the  original  and  in  Liouville's 
reprint,  is 

dY=-Z^tudt-Z^{\^  f)  du. 

The  incorrect  sign  in  the  second  member  has  been  corrected  in  the  reprint  in   Gauss, 
Werke,  vol.  IV,  and  in  the  translations. 

Art.  8,  p.  15,  1.  3.  Buler's  work  here  referred  to  is  found  in  Mem.  de  I'Acad. 
de  Berlin,  vol.  XVI,  1760. 

Art.  10,  p.  18,  11.  8,  9,  10.  Instead  of  2>,  D',  D"  as  here  defined,  the  ItaUan 
geometers  have  introduced  magnitudes  denoted  by  the  same  letters  and  equal,  in 
Gauss's  notation,  to 

D  D'  D" 

V{EG-F^)'     V{UG-Fy     V{EG-F^) 
respectively. 

Art.  11,  p.  19,  U.  4,  6,  fr.  bot.  In  the  original  and  in  Liouville's  reprint,  two  of 
these  formulae  are  incorrectly  given  : 

dF         „,  dF      1    dE 

=  m    -r  n,    n  = 

dq  dq        2     dq 

The  proper  corrections   have  been  made  in  Gauss,  Werke,  vol.  IV,  and  in  the  trans- 
lations. 

Art.  13,  p.  21,  1.  20.  Gauss  published  nothing  further  on  the  properties  of  devel- 
opable surfaces. 


54  NOTES 

Art.  14,  p.  22,  1.  8.  The  transformation  is  easily  made  by  means  of  integration 
by  parts. 

Art.  17,  p.  25.  If  we  go  from  the  point  p,  q  to  the  point  {p  +  dp,  q),  and  if  the 
Cartesian  coordinates  of  the  first  point  are  x,  y,  s,  and  of  the  second  x  +  dx,  y  +  dy, 
s  +  ds;  with  ds  the  linear  element  between  the  two  points,  then  the  direction  cosines 
of  Js  are 

dx  n     dy  ds 

cosa=:— -,     cosB=:-~,     cos  ■)/  =  --. 
ds  ds  ds 

Since  we  assume  here  §'=  Constant  or  dq=^0,  we  have  also 

^^^a^'^^'     ^y^Yp'^P^     ^^^d^'^P'     ds  =  ±  \/E.dp. 

If  dp  is  positive,  the   change  ds  will  be  taken  in  the  positive  direction.      Therefore 
ds  =  V  E .  dp, 

1       dx  1       dy  1       ds 

In   like  manner,  along   the  line  p  =  Constant,  if  cos  a',  cos  /8',  cos  y'  are  the  direction 
cosines,  we  obtain 

1       dx  /,,         1       2^  Ids 

cosa=-pr^-^,     cos^==-j7^-3^,     cosy=^7^-g^. 

And  since 

cos  ftj  =  cos  a  cos  a'  +  COS  /8  cos  yS'  +  cos  y  cos  y', 
F 

From  this  follows 

V{EG-F^) 

'''''"-" — Veg — 

And  the  area  of  the  quadrilateral  formed  by  the  lines  p,  p  -^  dp,  q,  q  +  dq  is 

dcr^V{EG-F').dp.dq. 

Art.  21,  p.  33,  1.  12.  In  the  original,  in  Liouville's  reprint,  in  the  two  French 
translations,  and  in  Boklen's  translation,  the  next  to  the  last  formula  of  this  article 
is  written 

E^8-F{a8+I3y)+Gay  =  -§^^^^.F' 


NOTES 


55 


The  proper  correction  in  sign  has  been  made  in  Gauss,  Werke,  vol.  IV,  and  in  Wan- 
gerin's  translation. 

Art.  23,  p.  35,  1.  13  fr.  hot.  In  the  Latin  texts  and  in  Roger's  and  Boklen's 
translations  this  formula  has  a  minus  sign  on  the  right  hand  side.  The  correction  in 
sign  has  been  made  in  Abadie's  and  Wangerin's  translations. 

Art.  23,  p.  35.  The  figure  below  represents  the  lines  and  angles  mentioned  in 
this  and  the  following  articles  : 


Art.  24,  p.  36.      Derivation  of  formula  [1]. 

Let  r^  =/  +  (f  +  E^  + E^+ E^  + R^+  etc. 

where  R^  is  the  aggregate  of  all  the  terms  of  the  third  degree  in  p  and  q,  i?^  of  all 
the  terms  of  the  fourth  degree,  etc.  Then  by  differentiating,  squaring,  and  omitting 
terms  above  the  sixth  degree,  we  obtain 


dp 


dp. 


d Re        ,     3j?o       _9-S,  dR,       ^d R„  dR 
+  2^r^  -~+  2 


and 


9/ 
dp     '    ^^  dp     '    "  dp     dp 


dp 
)R_ 

dp     dp  ' 


dq 


dq 
+  4,^  +  4,^  +  2 


dq 
d Rr,  d R. 


dq 


dq     '  ^"^  dq     '   "  dq     dq  dq     dq 


56  NOTES 

Hence  we  haA^e 


dp        ^  dq  "       '^\dp'        *  \  dq  '       ^  dp    dp        ^  dq    dq 

./„  ^9^.\'  ,    , /3^.\'\   ,    ./.  „    ,    ,dR.dR,  ,      ai2,ai2A 

^*\^^6^^\a»/ ^Ma^ ;  ^  2  ajf,  a«  ^  2  a^  a<7 /' 


3jt?  '         4  \  a  g  /  /  \         5       "i   dp    dp 

3^  '        '^\dq  '        '^  dp    dp        "^  dq    dq 
since,  according  to  a  familiar  theorem  for  homogeneous  functions, 

d  R„  dR„       „   _„ 

^-97  + ^^^  =  3^-  etc. 

By  dividing  unity  by  the  square  of  the  value  of  n,  given  at  the  end  of  Art.  23,  and 
omitting  terms  above  the  fourth  degree,  we  have 

1  -  —  =-  2/°  q'  +  2f'pq^  +  2c/°q''-  if°-'q''  +  2f"pY  +  2ff'pq'^  +  2  h°  q\ 

This,  multiplied  by  the  last  equation  but  one  of  the  preceding  page,  on  rejecting  terms 
above  the  sixth  degree,  becomes 


(l  "  i)  (^)'=  ^f°P"l"  +  ^f'P"'!"         -  12./°y  ^*  +  8  h°fq^ 


dp 


+  8y°/^^        +8/'>V/    +2rf(^) 


dp. 
+  8/>^^^^  +  8//^^     +  8/y  q^^-^  +  8^>r/^^ 

Therefore,  since  from  the  fifth  equation  of  Art.  24 : 


NOTES  57 

we  have 

=  8/°/?'  +  8/'  /^'  +  8 g°ff  +  8/°jo^^y^  -  Vlffq^  +  8/"/^' 
+  8  //^^  +  8  A°//  +  2/°  ^^  f^)  V  8/y  ^^^^  +  8/>^^^^  +  8  g'^ p  /^. 
Whence,  by  the  method  of  undetermined  coefficients,  we  find 

R. = (I/"  -  i^nft + \9'f<t + (I  A°  -  iV/°')  /?*• 

And  therefore  we  have 

[1]  r'  =f  +  \rff  +  \fft  +  (I/"-  ^r')p'q'  +  etc. 

This  method  for  deriving  formula  [1]  is  taken  from  Wangerin. 
Art.  24,  p.  36.      Derivation  of  formula  [2]. 

By  taking  one-half  the  reciprocal  of  the  series  for  n  given  in  Art.  23,  p.  36,  we 
obtain 

^  =  i  [l-^o  q^  -f'pq'-g°  q'  -f'ff  -  g'pf  -  {h°  -/°^)  q'  -  etc.] . 
And  by  differentiating  formula  [1]   with  respect  to  p,  we  obtain 

^  =  2  [p  + 1/>  q^  +  y'p't  +  2  (I/"  -  ^nft 

+  (|A°-^/°^);.?*+etc.]. 
Therefore,  since 

1     ^{r") 
'•^^^'''  =  2^-"^' 

we  have,  by  multiplying  together  the  two  series  above, 

[2]  r  sin  >^^p-  \rpt  -  \f'p'f  -  a/"  +  i^nft  -  etc. 

-ka'ft-  \9'fq' 

-{ih°-i^npq\ 


68  NOTES 

Art.  24,  p.  37.      Derivation  of  formula  [3]. 

By  differentiating   [1]   on  page  57  with  respect  to  q,  we  find 

^  =  2  [^  + 1/°/^  +  ^f'p-q  +  (f /"  -  -hnp'q 

+  %9°P'9'  +  I///  +  (I  /^°  -  il/°^)/?^  +  etc.] . 
Therefore  we  have,  since 

[3]  r  cos ,/,  =  ^  +  |/>V  +  i/y  ^   +  d/"  -  A/°')/^  +  etc. 

Art.  24,  p.  37.      Derivation  of  formula  [4]. 

Since  r  cos  <^  becomes  equal  to  p  for  infinitely  small  values  of  p  and  q,  the  series 
for  r  cos  ^  must  begin  with  p.      Hence  we  set 

(1)  r  cos  <^  =p  +B^+B^  +  B.^+R,+  etc. 

Then,  by  differentiating,  we  obtain 

^  ^  dp  dp  dp  dp  op 

^^  3§'  dqdqdqdq 

By  dividing  [2]  p.  57  by  n  on  page  36,  we  obtain 

i^> / -  i///  -  (I /^°  -  ii/°')p  ?'• 


(4)  ^  =^  - 1/>^^  -  ^fp'q^  -  (!/"  +  A/°^)/?^  -  etc. 


Multiplying  (2)  by  (4),  we  have 

(5)  '-^■'-^-P^p'-^^/-^^/-^^/-^      -iif'^^npY 
-irpf-i/y/-^  -irp/-^-iff'ff 

-\f'ff     -  u'f/-^-{v^°  -iinp9' 


NOTES  59 

Multiplying  (3)  by  [3]  p.  58,  we  have 

^'  ^  dq  ^dq^^dq^^dq  ^  ^  dq  ^  "^J  P  ^  dq 

Since 

r  sin  i/»    9  {r  cos  <^)  d  {r  cos  <A) 

-^  ^p +  rcos,/,- g^— -^cos.^, 

we  have,  by  setting  (1)  equal  to  the  sum  of  (5)  and  (6), 
p  +i?2+i?3+i?4+ii'5+  etc. 

d R„    ,      dR„  dit.  .      9-ffc 

-p+p^+pj^  +p^      +p-3^  -iif^^-nnps' 


+  l/V?^  -|///  +  etc.. 


from  which  we  find 


R.-i^g'p'f  +  (f  ^°  -  A/°^)i'?*  +  (i^/"  -  A/°^)/?^- 

Therefore  we  have  finally 

[4]  r  cos  ^=p^  \rpf  +  3^/'/?^  +  {-i^r  - i^J°^)ft  +  etc. 

+  hfvi^  to//?' 

Art.  24,  p.  37.      Derivation  of  formula  [5]. 

Again,  since  r  sin  ^  becomes    equal   to    q  for   infinitely  small   values   of  p  and  q, 
we  set 

(1)  /•sin<^  =  ^+i?2+i?3  +  i?4+i?6+ etc. 


60  NOTES 

Then  we  have  by  differentiation 

d{rsmcf>)  __dR       dR       dR,      dR, 

(^)  o o r  -5 1-  -^^  +  -r—  +  etc. 

^  '  dp  dp  dp  dp  dp 

^  ^  Sj-  dj"         dj'  dq         dq 

Multiplying  (4)  p.  58  by  this  (2),  we  obtain 

^   ^  n  dp  ^  dp       ^  dp       ^  op  ^  dp  •*•'    ^  *    3j» 

3/  i'?  ajo     ^J  P^  dp     ^3  P^  dp     ®^^- 

Likewise  from  (3)  and  [3]  p.  58,  we  obtain 

(5)  .cos,.^i^)=,+,^^+,^+,^+,^^       +(ir-A/-)/. 

+  l/Vi?  +  l/°/?^'  +  l/°/?  ^+  I//?' 


+  |yV?'  +  I^V?'^'+etc. 


Since 


dq 
R 


rsin»/»   9  (r  sin  (6)  3(rsin<i)  .     . 

^      ^         ^^4-rcosi|»--^ — ^  =  rsin<^, 


m  3^  ^  dq 

by  setting  (1)  equal  to  the  sum  of  (4)  and  (5),  we  have 
q  +i?2  +  i?3+i?,+J?g+  etc. 

+.^^+i/y.  +.^^  +  i/y,^^+.^^  +  etc. 


,         9^2 

,      9^2 
^  3y 

^    3§' 

■\g°ff-  i^9'f<i 


NOTES  61 

from  which  we  find 

R.= -  (ivr  -  -hnp'q  -f,9'f<t  -  (i  h<^^^nfq\ 

Therefore,  substituting  these  values  in  (1),  we  have 

[5]  r^m^  =  q-\rfq-\ffq    -  (J^/"-^/°^)/^-etc. 

Art.  24,  p.  38.     Derivation  of  formula  [6]. 
Differentiating  n  on  page  36  with  respect  to  §>,  we  obtain 

(1)  a^  =  2/°  ^  +  2/>  q  +  If'fq  +  etc. 

Zg°q'  ^-^g'pf  +  etc. 

+  4  A°  ^^    +  etc.,  etc. 
and  hence,  multiplying  this  series  by  (4)  on  page  58,  we  find 

T  Sin  \\i    3  f? 

(2)  —^  .-  =  2rpq  +  2ffq  +  2f"fq  +  3 /ff  +  etc. 

+  Sff°pq'+{4:h°-ir')pq\ 

TT 

For  infinitely  small  values  oi  r,  \\i -\-  <^  =  ^,  a.s,  is  evident  from  the  figure  on  page 
55.    Hence  we  set 

»|<  +  <^  =1  +i?i  +  ii;2+i?3+i?4+  etc. 
Then  we  shall  have,  by  differentiation, 

^   '  dp  dp  dp  dp  dp 

^  '  dq  dq         dq         dq         dq 

Therefore,  multiplying  (4)  on  page  58  by  (3),  we  find 

rsmy\,   d{x\,  +  <f>)^    ^1  .     ^2  ^    ^3  ^    ^  +  etc 

^   '  n  dp  P  dp         "  dp         "  dp  ^    dp 

-irpZ-^-irp/-^ 


62  NOTES 

and,  multiplying   [3]  on  page  58  by  (4),  we  find 


And  since 


rsini/»    9w       rsin»/(   3(i//  +  <i)  3(i//+<i) 

5 1 5 h  r  cos  \I» :; — '—  =  0, 

n         dq  n  dp  ^         dq  ' 


we  shaU  have,  by  adding  (2),  (5),  and  (6), 

^=^^^  +  2/>^  +  2/y^         +2rfq  _3^o^,^3^ 

'  From  this  equation  we  find 

i?i=0,     R=-f°pq,     R^=-\f  p^q-(ffq\ 

R.=-  a/"  -  \n  /  ?  - 1  ^y  q'  -  (/*°  -  i/°^)  j^  ?^- 

Therefore  we  have  finally 


[6]  '^ + (^ = 2  -f°p'i  -y'fi  -  (if"  -  \np"9  -  etc. 


■9°pf-  \9'f<t 


NOTES  63 

Art.  24,  p.  38,  1.  19.  The  clifFerential  equation  from  which  formula  [7]  follows 
is  derived  in  the  following  manner.  In  the  figure  oh  page  55,  prolong  AD  to  D', 
making  DD'  =^dp,  through  D'  perpendicular  to  AD'  draw  a  geodesic  line,  which  will 
cut  AB  in  B'.  Finally,  take  D'B"  =  DB,  so  that  BB"  is  perpendicular  to  B' D' . 
Then,  if  by  ABD  we  mean  the  area  of  the  triangle  ABD, 

dS      .    AB'D'-ABD      ,.    BDD'B'       .    BDD'B"   ,.    DD' 
g^  =  hm ^^, =  hm     ^^,      =hm     ^^,      •^^^' 

since    the   surface  BDD'B"   differs  from   BDD'B'   only  by  an   infinitesimal   of  the 
second  order.     And  since 

r  BDD' B"       r  I  -^       Xp 

BDD'B"  =  dp  .j  ndq,  or  Mm. — jj^, —  ^Xndq,  (jA    '' 

and  since,  further, 

,.     DD'      dp 
^BW  =  d^' 


consequently 


Therefore  also 


dS_dp  ^ 

dr       dr  J        "' 

dS   dp       dS   dq       cip     r 
do    dr       dr  J 


dp    dr       dq    dr 

dr    dr 
Finally,  from  the  values  for  r— ?  z—  given  at  the  beginning  of  Art.  24,  p.  36,  we  have 

dp      1   .  dq 

r-  =  -  sin  ti,     5—  =  cos  i/», 
dr      n       "      or  "' 

so  that  we  have 

dS  sin  \\i      dS  sin  tA    r 

= h  5—  •  cos  ti  = I  n  dq. 

dp       n         dq  ^         n      '^        ^ 

[Wangerin.] 

Art.  24,  p.  38.      Derivation  of  formula   [7]. 

For  infinitely  small  values  of  p  and  q,  the  area  of  the  triangle  ABC  becomes 
equal  to  \  pq-  The  series  for  this  area,  which  is  denoted  by  S,  must  therefore  begin 
with  ^  pq,  or  R^.     Hence  we  put 

S^R^  +i?3  +  i?,+i?6+i2g+etc. 


^^ 


64  NOTES 

By  differentiating,  we  obtain 

^   ^  dp         dp  dp  dp  dp  dp  *' 

^    '  d  q        dq         d q         d  q         d  q         d  q  *' 

and  therefore,  by  multiplying  (4)  on  page  58  by  (1),  we  obtain 

rsinxf,   dS_    d_R^        dR^        d_R^  ,      ^  aig^ 

^  '  n         dp      P  dp       P  dp       P  dp  P  dp  "dp 

dR„  „d  R,  „dR. 

-irr<f^-irp<f-sf-iri"f^ 

^        o  3  ^^2  Q        o  .'^^^ 


dp       "^^  i"i    dp 

7       /      2     3^-^2 

9  7? 
and  multiplying   [3]   on  page  58  by  (2),  we  obtain 

.  ^^      3^2  ,    3^3  ,    3i2,  ,    dR.  ,    ai?g 

(4)        rcos^.3-  =  ^^  +  ^^  +  ^^  -^q-^  +?^^  +  etc. 

P       T       /       3      2  ^-^2 


NOTES  65 

Integrating  n  on  page  36  with  respect  to  q,  we  find 

(5)  jndq  =  q  +  \r  f  +  i/>?'+  i/'y  t  +  etc. 

+  iy°?*  +T/i^?*+  etc. 

+  ^  A°  §'^  +  etc.  etc. 

Multiplying  (4)  on  page  58  by  (5),  we  find 

(6)  '^  -Sn  dq  =pq-fpq-  -  ^f'fq-  -  (i|/"  +  -^nff  -  etc. 

Since 


rsinii   9^'  3>S'      rsinii*    r 

-li — V  r  cos  «|(  •  r—  = -I  ndq. 

n        op  ^    dq  w       *'        ^ 


3p  ^    d  q 

we  obtain,  by  setting  (6)  equal  to  the  sum  of  (3)  and  (4), 

pq  -fpf  -Wp'^f        -(iir  +  ^/°')/?^-etc. 

—  \9°Pt        —  If//?* 

=^V"^^V^^V         +^-a7+^^  +  ^^         +f^V^^^+etc. 

-^/^+^^+^^^      -i/>.^^^+.^^      -m"^i.nf/-^ 
+ 1/°/  ?^^  -  i/y  .^^^ + 1/°/ .  ^^-  i//^^  ^^ 

From  this  equation  we  find 

R,=  \pq,     i?3=0,      E,^--^rpq'-^rp'q, 

R.= -  ^f'p'q  -  ^ff°p'f  -  Thfp'f  -^9°p  q\ 
-  ^g'p'q'  -  i-hf"  -  i^nfq  -^9'fq'- 


66  NOTES 

Therefore  we  have 

[7]  S=\pq-i^rpq^-i^fp'q      -  {^f  -  ^fo^)f  q  -  etc. 

Art.  25,  p.  39, 1.  17.  3  /  +  4  ^^  +  4  ^  ^'  +  4  ^'Ms  replaced  by  3/  +  4  ^^  +  4  q'"". 
This  error  appears  in  all  the  reprints  and  translations  (except  Wangerin's). 

Art.  25,  p.  40,  1.  8.  ^p^  —  lf^qq'^Aiq  q'  is  replaced  by  "&  jj^  —  2  q^  +  qq' 
+  4  q''^.      This  correction  is  noted  in  all  the  translations,  and  in  Liouville's  reprint. 

Art.  25,  p.  40.      Derivation  of  formulES   [8],   [9],   [10]. 

By  priming  the  §''s  in  [7]  we  obtain  at  once  a  series  for  S' .  Then,  since 
(T=^S—S',  we  have 

T  =  \p{q-q')-i^rp'{q~q')    -  ^f  /  {q  -  q')     -  ^^  9°  p' {q' '  q") 

-^rp  {q'-q")-Thf'fiq'-q'')-^  rp  {q'-q'% 

correct  to  terms  of  the  sixth  degree. 

This  expression  may  be  written  as  follows  : 

cr  =  ^p  {q-q')  (1-  \  /°  (/  -\- q' +  q  q' +  q") 

-^f'p{Qp''+7q''+7qq'  +  7q'^) 

-^9°iq  +  q')i^p'  +^q'  +^q")), 

or,  after  factoring, 

(1)      <r^^p{q-q')0-irf-if'pf-^g°q')(l-ir{p'-q'  +  qq'  +  q'') 

~-^fp{Qp'-Sq''+7qq'+7q")-^ff°{3p'q  +  SpY-Qq''  +  4:q'q'  +  Aqq"  +  4tq")). 

The  last  factor  on  the  right  in  (1)  can  be  written, thus  :   __.-  ;  ") 

V--Thr{y)  -Tfo/°&/)  -Thf'p{Qqq')-Thr\h')    -Thfpiqq') 

+  Thf°i^f)  +Thr{^f)  -Thf'p{^q")+Thri^q')    -Thf'pi'^q") 

-Thr{^qq')~Thr{Qqq')-Thff°q{^f)  -rhnqq')    -rhrq'i^p') 

-Thr  i^q")  -ihr  i^q")  +Th9°9i^f)  -Thri^q'^)  +Th9°q'{^q') 

-Thfp{^f)-Th9°q{^qq')-Thf'p{^f)  -Thff°q'iqq') 

+  Thf'piQq')-Thf9°q{^q"')  +tIt/>(2/)  -tIt7^Y(%")). 

We  know,  further,  that 


NOTES  67 

/=/° +/>+/"/+ etc., 
g  =  g°  +  g' p  +  g" p^  +  etc.,    ' 
h=h°  +  h'p+  h"p'+  etc. 

Hence,   substituting  these   values   for  /,  g,  and   h  in  k,  we  have  at  B  where  k  —  fi, 
correct  to  terms  of  the  third  degree, 

/3  =  -  2/°  -  2/>  -Qg°q-  2f"  p'' -  &  g'  p  q- {12  h°  -  2/°')  f. 

Likewise,  remembering  that  q  becomes  q'  at  C,  and  that  both  p  and  q  vanish  at  A, 
we  have 

y  =  -  2/°  -  2/>  -^g'^q'  —  2/"/  -  6  g' p  q'  —  (12  h°  —  2/°^)  q'% 

a  =  -2f°. 

And  since  c  sin  B  —  r  sin  xfi, 

c  sin  B=p{l-  ^f°  f  -  i/>  f  -  iy°  ^^  -  etc.).  V  ' 

Now,  if  we  substitute  in  (1)  c  sin  5,  a,  ^,  -y  for  the  series  which  they  represent, 
and  a  for  5*  —  cf ,  we  obtain  (still  correct  to  terms  of  the  sixth  degree) 

(r  =  i«c  sin5  (1  +  xk  a  (4/-  2  ^^+  3  ^^'  +  3  q'") 
+  Tk/8(3/-6^^+6^?'+3/^) 

And  if  in  this   equation  we  replace  p,  q,  q'  by  c  sin  5,  c  cos  5,  c  cos  B  —  a,  respect 


^U' 


— —^~^i 


ively,  we  shall  have  ' '         <     ^ 

[8]  o-  =  iacsin5(l  + Y^o  a(3«''+4c2—    9  accost)  '^^     ^ 

+  Ti(r/8(3a'+  3  c^- 12  ac  cos5) 

+  TTO  r(4«'+ 3c^-    9  accost)). 

By  writing  for  B,  a,  /8,  a  in  [8],  A,  /3,  a,  5  respectively,  we  obtain  at  once 
formula  [9].  Likewise  by  writing  for  B,  fi,  y,  c  in  [8],  C,  y,  /8,  b  respectively,  we 
obtain  formula  [10].  Formulae  [9]  and  [10]  can,  of  course,  also  be  derived  by  the 
method  used  to  derive   [8]. 

Art.  26,  p.  41,  1.  11.  The  right  hand  side  of  this  equation  should  have  the  pos- 
itive sign.     All  the  editions  prior  to  Wangerin's  have  the  incorrect  sign. 

Art.  26,  p.  41.      Derivation  of  formula   [11]. 

We  have 
(1)  r'^  +  r'^—  {q  —  q'f  —  2r  cos  <^  .  r'  cos  ^'  —  2  r  sin  ^  .  r'  sin  ^' 

^h"^  <^-a^-2bc  cos  (<^  -  f ) 
=  2  he  (cos  A*  —  cos  A), 
since  h"^  +  c^  —  a^  =  2  b  e  cos  A*  and  cos  (^  —  ^')  =  cos  A. 


68  NOTES 

By  priming  the  g's   in   formuke   [1],    [4],    [5]    we    obtain  at  once  series  for  r''^, 
r'  cos  <^',  /  sin  0'.     Hence  we  have  series  for  all  the  terms  in  the  above  expression, 
and  also  for  the  terms  in  the  expression : 

(2)  r  sin  (^  .  r'  cos  <^'  —  r  cos  ^  .  r'  sin  (j)'  =  be  sin  A, 
namely, 

(3)  r'  =  /  +  |/°  /  q'  +  i/'/  q'  +  (f /"  -  A/°')/  f  +  etc 

(4)  r"==p'+if°p'q"+yYq"  +  {lf'-^f°')pV'  +  etc. 

(5)  -{q-qy  =  -q'-V2qq'-q'\ 

(6)  2  r  cos  (^=2^  +  1/°^;^^  +  J4/';//+  (T^/"-i|/°^)//  +  etc. 

^g°pf     +\^g'fq'' 

(7)  r'  cos  f  =;;  +  |/°y>  q"^  +  -rV'r ?'■'  +  (fo/"  -  i^r)f€'  +  etc. 

(8)  2  r  sin  (^  =  2  ^  -  f /°  /  q  -  |/'/  ^   -  {-^f  -  ^4/°^)  /  q  -  etc. 

(9)  r'  sin  f  =  ^'  -  i/°  fq'  -  \f'fq'    -(yV/"  "  9^/°'")/?'  "  etc 

By  adding  (3),  (4),  and  (5),  we  obtain 

+  2  qq'  +i^y (f +  /')  +  |/i«''  (?'+  /^) 

On  multiplying  (6)  by  (7),  we  obtain 
(11)  2  r  cos  <^  .  r'  cos  <^' 

= 2  / + %rf  (/ + /2)  +  f /y  (^^ + ^'^)  +  (I/"  -  ^r)f  a + /^)  +  etc. 


NOTES  69 

and  multiplying  (8)  by  (9),  we  obtain 

(12)  2  r  sin  4>  .  r'  sin  ^' 

=^2qq'-  %rfqq'  -  if'fqq'  -(!/"  "H/"')/?/ -  etc. 

-\g°  fqq'  {q^  q')  -  ^ 9' p" qq'  {q  +  q') 

Hence  by  adding  (11)  and  (12),  we  have 

(13)  2Jccos^ 

=  2  /  +  1/°/  {q'  +  q")  +  ifp'  (5  f~  4  qq'  +  5lq'')-^r'f{2  q'  +  2q''-  3  qq')  +  etc. 

+  2qq'-^f°p^qq'  +  ^  ffY  {2  q'+  2  q"-  q'q' -qq") 

-  J^/°2/  (14  ^*  +  14  ^'*  +  13^^  ^'  +  13  ^^'*  -  40  q^q'^) 

+  iV^y  (7^+7  q"  -Bq'q'-S  qq") 

+  if"pHSq'+^q"-2qq') 

+  I  h°f  {2q^+2q"-q^q'-  qq'^). 

Therefore  we  have,  by  subtracting  (13)  from  (10), 

2  he  (cos  J.*  —  cos  A) 

—  \rf  iq'  +  q"-'^qq')~\f'p'  {q'  +  ?''  -  2  qq')+  ^rf{<t  +  q"-2  qq')  -  etc. 
-  ^(/°p\q^  +  q'^-  tq'  -  qq''-)-\f"  f{<t  ^q'^-2qq') 
+  T5/°V(7??*+  7/*+  13/^'+  13^/^-40^^^'^) 
-  I  h°f  {c/  +  q"-q^q'-qq'^) 

-Toff'p  {q'  +  q"-q'q'-qq")> 

which  we  can  write  thus  : 

(14)  2  be  (cos  A^  -cos  A)  =-2  /  (ry  -  q'f  (l/°  +  i/>  +  i^°  {q  +  q')  +  ^o/"  / 

+  \h°  {q'^  qq'+  q'')  +  ^9'p{q  +  q') 
-l25/°V-9V/°M7r+7/^+27^^')), 

correct  to  terms  of  the  seventh  degree. 

If  we  multiply  (7)  by   [5]   on  page  37,  we  obtain 

(15)  r  sin  <^  .  r'  cos  ^' 

=pq  +  irpqq"+f^f'p'qq"+{-hf"-f,np'qq"-^^^- 

-ipp^q    +  h9°pqq'''  +  ^^g'p'^qq'^ 

-\9°ff    -y°'p'qq" 

-^g'p^q^ 
-{\h°-^\^np'q\ 


70  NOTES 

And  multiplying  (9)  by  formula  [4]   on  page  37,  we  obtain 

(16)  r  cos(f>  .  r'  sin  <^' 

=pq'-krp'9'  -i//?'     -(■iVr-9V/°'')//+etc. 

+  irpq'q'  -\g^fq"      -  ^^ 9' p' q" 

+  h9°pq'q'  -^fffq' 

+  {-hf"-i^np'q'q' 
+  -hd'p^q'q' 
^i.\^°-iznptq'- 

Therefore  "we  have,  by  subtracting  (16)  from  (15), 

(17)  icsin^ 

-p{q-  q')  (1  -  i/°/  -  ^f'pqq'        -  i^f'-^np' qq' 
- 1/°  q  q'  -  iff  -  i-hf"  -  ^nf 

-^ff°qq'{q  +  q')  -  -hg'p  qq'{q  +  q') 
-i9°p'{q  +  q')  -^ff'p'iq  +  q') 

-{ifi°  +  unp'iq'+qq'  +  q") 

-{ih°-^r')qq'{q'+qq'+q") 

+  U°'fqq% 

correct  to  terms  of  the  seventh  degree. 

Let  A*—A  =  C,  whence  A*=^A+t„  I,  being   a   magnitude  of  the   second   order. 
Hence  we  have,  expanding  sin  t,  and  cos  ^,  and  rejecting  powers  of  £,  above  the  second, 

cos  A*— cos  A  .  (l  — 2")  ~  sin^  •  Cj 
or 

cos  A*-  —  cos  4  = 2 C,   —  smA  .  Q; 

or,  multiplying  both  members  of  this  equation  by  2  b  c, 

(18)  2  b  c  (cos  il*  —  cos  ^)  =  —  ^  c  cos  ^  .  ^'  —  2  J  e  sin  A  .  ^ 

Further,  let  ^  =i?2+^3+-^4+etc.,  where  the  B's  have  the  same  meaning  as  before. 
If  now  we  substitute  in  (18)  for  its  various  terms  the  series  derived  above,  we  shall 
have,  on  rejecting  terms  above  the  sixth  degree, 

(/  +  qq')  ^/+  ^p  iq  -  q')  (i  -  i/°  {f  +  2  qq'))  in^+B.+E,) 

=  2/  {q-q'f  (i/°  +  if'p  +  i^°  {q  +  q')  +  ^f" p'  +  -i^g'p  {q  +  q') 
+  \h°{q'+qq'+q'')--hr'{l2p-+1q'^-1q''+21qq')y 


NOTES  71 

Equating  terms  of  like  powers,  and  solving  for  R^,  B^,  R^,  we  find 

R^=p{q- q')  .  i/°,     i?3  =p  [q  - q')  (i/>  +  \g°  {q  +  q')), 
Ji,=P{g-q')i-^rf+-ioff'pi9  +  g')  +  if^°  iq'+qq'+q'') 

Therefore  we  have 

A^-A=p{q-q') (i/°  +  i/>  +  iff°  {q  +  q')  +  J^/"/ 
+  -ha'P  {1  +  ?')  +  i^°  it  +  11'  +  ?") 

correct,  terms  of  the  fifth  degree. 

This  equation  may  be  written  as  follows : 

+  iVr  /  +  -ha'p  (^  +  ?')  +  i>^°  (?'+  11'^  ?'^)  -  9V/°'(2/+  2?^+  7^^'+  2^-)). 
But,  since 

2  o-  =  «jj  (1  -  i/°  (/  ^cl^qq'-\-  q'^)  +  etc.), 
the  above  equation  becomes 

A*^A-ai-ir-  y'p  -  \g^  {q  +  q')  -  \f"  p'  -  ^  g' p  {q  +  q') 

-^h°  (/  +  qq'  +  q")  +  ^f°'  (4/  +4:q^+Uqq'+i  q")), 
or 

A*=A-cri-ir-^r    -A/° 

-Ti9°1    -T2  9°2' 

-  -hg'p  1  -  -hs'f  i'  +  \g'v  {i  + 1') 

-\^h°q'  -\^h°  q'^  +  ^h°{Sq^-2qq'  +  Sq'^) 
+  A/°'  l'  +  ^r'r  +  9^/°'  (4/  -llq'+  14  ^/-  11  ^'^)). 
Therefore,  if  we  substitute  in  this   equation  a,  ^S,  y  for  the  series  which  they  repre- 
sent, we  shall  have 

[11]  A*  =  A-cri\a  +  i^fi  +  1^7  +  ^f"p'+\9'p{l  +  l') 

+  ^h°{2>q^-2qq'+^q'^)  +  -^f°'  (4/  -  11  ^^  +  14  qq'  -  11  q'')). 

Art.  26,  p.  41.      Derivation  of  formula   [12]. 

We  form  the  expressions  {q  —  q'f  +  i^  —  r''^  —  2{q  —  q')  r  cos  \|»  and  {q  —  q')  r  sin »/». 
Then,  since 

{q- q'Y  -^  r"  —  r'""  =  a^  +  c^  —  h"  ^  2  ac  cos  B*, 

2  (y  —  q')  r  cos  }fi  =  2  ac  cos  jB, 


72  l^OTES 

we  have 

{q  —  qy  -|-  r"  —  r'^—  2  (q  —  q')  r  cos  t/;  =  2  a  c  (cos  5*  —  cos  5). 

We  have  also 

{q  —  q')  r  sin  xjt  ^^  ac  sin  -B. 

Subtracting  (4)   on  page   68  from   [1]    on  page  36,  and  adding  this   difference  to 
{q  —  q'Y,  we  obtain 

(1)  {q  -  q'f  +  r^  -  r'\  or  2  ac  cos  5* 

=  2  <?  (^  -  /)  +  |/°/(^^  -  q'^)  +  i/y  (^^  -  q'')   +  (I/"  -  ^/°^)/  (^^  -  ?'^)  +  etc. 

If  we  multiply   [3]   on  page  37  by  2{q  —  q'),  we  obtain 

(2)  2{q  —  q')  r  cos  t/;,  or  2  ac  cos  B 

=  2^  (^-/)  +  l/y^(^-/)  +/y^(?-^')     +  (!/"- A/°'")/^  iq-2')  +  etc. 

-K|/^°-|f/°')//(?-?')- 

Subtracting  (2)  from  (1),  we  have 

(3)  2  «c  (cos  5* -cos  5) 

=  -2p'{q-q'fi\r  +  \f'P  +  (ir- A/°')/+  etc. 

Multiplying  [2]   on  page  36  by  {q  —  q'),  we  obtain  at  once 

(4)  {q  —  q')  r  sin  i/;,  or  a  c  sin  B 

=  p{q-  q')  (1  -  i/°  ^^  -  i/>  q^  -  {}/"  +  ^D  ff  +  etc. 

We  now  set  B-^—B  =  l,,  whence  B*=B+C,  and  therefore 

cos  5*  =  cos  B  cos  ^  —  sin  5  sin  ^. 

This  becomes,  after  expanding  cos  ^  and  sin  ^  and  neglecting  powers   of  £,  above  the 
second, 

cosB      „ 
cos  jB*  —  cos  B  = i^ —  •  ^ —  sin  B .  4. 

Multiplying  both  members  of  this  equation  by  2  ac,  we  obtain 

(5j  2  ac(cos^'''  —  cos  B)  =  —  ac  cos  B  .  t^  ~  2  ac  sin  B  .  4. 


NOTES  73 

Again,  let  t,=R^+R^-\-R^-{- Qic,  where  the  i?'s  have  the  same  meaning  as  before. 
Hence,  replacing  the  terms  in  (5)  by  the  proper  series  and  neglecting  terms  above  the 
sixth  degree,  we  have 

(6)  q{q-q')R,^  +  2p{q-q'){l-\rq'){R,+  R,-^R,) 

=  2/  {q  -  qj  i\r  +  i/>  +  {\f"  -  ^np' 

+  \g°{2q-\-q')-^^g'p{2,q  +  q') 

+  {\h°--J^n{^q'+2qq'+q'^)y 

From  this  equation  we  find 

R.=p {q  -  q')  ■  i/%   R.=p  (?  -  2')  (i/>  +  i^°  (2  ?  + 1')), 

R,=-p{q-q')iif"p'+^ff'p{2q  +  q')+ih°{^q'+2qq'+q'') 
-^r\ip'+lQq'+9n'+7q'')y 

Therefore  we  have,  correct  to  terms  of  the  fifth  degree, 

B*-B=p{q-  q')  (i/°  +  \f'p  +  i/"/ +  ^g'p  (2  q  +  q') 

+  \g°{2q  +  q')  +  \h'^(3q'+2qq'^-q''') 

or,  after  factoring  the  last  factor  on  the  right, 

(7)  B-^'=B-^p{q-q'){l-\r{p'+q'^+qq'+q''))i~y°-\f'p-\g°{2q^q') 

+  To/°'(-2/+22^^+8^^'  +  4^'^)). 
The  last  factor  on  the  right  in  (7)  may  be  put  in  the  form : 

—  _2_  fo  —   1  -fo  —  _2_  /o 

12/  6/  12/ 

—  ^g°q    -~^g°q' 
-%f"f  -^f'f  +1^/"/ 

—  %g'pq  -^g'l^q' +  ^g'p{^q  +  q') 

-l^h°  q"  -  \^h°q'^  ^  \h°  {4.q'+  Z  q'^-4.qq') 

Finally,  substituting  in  (7)  a,  a,  /3,  y  for  the  expressions  which  they  represent,  we 
obtain,  still  correct  to  terms  of  the  fifth  degree, 

[12]  B-^=B-a(i^a+ifi  +  ^y  +  ^f"f 

+  -^g'p{2q  +  q')  +  ih°{4:q'-Aqq'  +  Sq") 


74  NOTES 

Art.  26,  p.  41.      Derivation  of  formula   [13]. 

Here    we    forra    the    expressions    {q  —  q'f  +  i-'"^  —  r^— 2  {q  —  q')r' cos  {it  — xji')  and 
{q  —  q')  r' sin  (tt  —  xji')  and  expand  them  into  series.      Since 

{q-q'f+  r"-f^=  0,'+  P-c''=  2  ab  cos  C*, 
2{q  —  q')r'  cos  {TT  —  xf,')  =  2ab  cos  C, 
we  have 

{q  -  q'Y  +  r'^-r'-2{q-  q')  r'  cos  {tt~xI,')  =  2  ab  (cos  0*  -  cos  C). 
We  have  also 

(q  —  5'')  r'  sin  (tt  — 1/(')  =  a  5  sin  C. 

Subtracting  (3)  on  page  68  from  (4)  on  the  same  page,  and   adding  the  result  to 
(q  —  q'Y,  we  find 

(1)  {q  -  q'f  +r"- r\  ox  2  ab  cos  C* 

=  -2q'{q-<^)-irf{f-q'-)-\J'f{f-q'-)    -  (2_^"-^/o2)^*(^2_  ^,2)  _  .^c. 

-\g''f{f-q")-\g'f\f-<^') 

By  priming  the   ^-'s  in  formula  [3]    on  page   37,   we  get   a    series   for  /  cos  i|(',  or  for 
—  r' cos  (tt  — x/;').      If  we  multiply  this  series  for  — /•'cos(7r  — 1/»')  by  2(^  —  q'),  we  find 

(2)  —  2(^  — ^')/cos(7r  — f),  or  —  2a5cos(7 

=  2{q-  q')  iq'  +  ^f^p^q'  +  \ffq'     +  (|/"  -  ^/°^)/^'  +  etc. 

And  therefore,  by  adding   (1)  and  (2),  we  obtain 

(3)  2a5(cosC*-cosC) 

=  -  2/  {q  -  q'f  (i/°  +  i/>  +  a/"-  ^/°^)/+  etc. 

+  ir(^  +  2^')  +  i/i'(?  +  2?') 

+  a/'°-9V/°')(?^+2^/+3/^)). 

By  priming  the   ^''s  in   [2]    on   page    36,  we   obtain   a   series  for  /  sin  »|/',  or  for 
r' sin  (tt  — »//').      Then,  multiplying  this  series  for  /  sin  (tt  —  i//')  by  {q  —  q'),  we  find 

(4)  {q  —  q')  r'  sin  (tt  —  \\i'),  ox  ab  sin  C 

=  p(q-  q')  (1  -  i/°  q"-\f'pq"-  (i/"+  ^r')fq"-  etc. 


As  before,  let  C*  —  C  =  ^,  whence  C*  —  C  +  ^,  and  therefore 
cos  C*=  cos  Ccos  ^  —  sin  Csin  £,. 


NOTES  75 

Expanding  cos  C  and  sin  ^  and  neglecting  powers  of  ^  above  the  second,  this  equation 
becomes 

cos  C*  —  cos  C= 2 C—  sin  C.  t„ 

or,  after  multiplying  both  members  by  2  ab, 

(5)  2«5(cosC*-cosC)=-«$cosC.  C—^.ab&hiC.  t,. 

Again  we  put  i^='R^  +  R^-\- R^-^  Qia.,  the  R&  having  the  same  meaning  as  before. 
Now,  by  substituting  (2),  (3),  (4)  in  (5),  and  omitting  terms  above  the  sixth  degree, 
we  obtain 

q'{q-q')Ri-2p{q-q'){l-\rq'^){R^  +  R^  +  R:^ 

+  iy°(?  + 2 /)  +  !//>(? +2^0 

+  {\h°-i^n{q'+2qq'+^q'-)), 
from  which  we  find 

R,=  p{q~q').\r,     R,=p{q-q')iU'p-^\9°{q  +  ^q')'), 
R,-p[q-q')i\f"p'+\g'p{q  +  2q')+\h°{q'+2qq'+^q'^) 

Therefore  we  have,  correct  to  terms  of  the  fifth  degree, 

(6)  C*  -C=p{q-  q')  (i/°  +  i/>  +  i/y  +  \g'p  {q  +  2  q') 

+  \g°  (?  +  2  ?')  +  I  h°  (?^  +  2  ?/  +  3  q"') 

-  9lo/°'(4/+  1q^  +^qq'  +  16  q'^)). 

The  last  factor  on  the  right  in  (6)  may  be  written  as  the  product  of  two  factors,  one 
of  which  is  \{}-—\f°[p'^-\-q'^-^qq'+q''^)),  and  the  other, 

2  a/° + i/>  +  iy°  (? + 2  /)  +  i/'y + i^y  ? + 2  ?o 

+  \h°{q''+2,  q'^  +  2  ?/)  -  gi^/°==(-/  +  2  ?^  +  4  ??'  +  11  ?'^)), 
or,  in  another  form, 

—  /■  2/0  2/0  2  /o 

V         T^/  T2"/  "g'/ 

--hf'P       -y'P 

--hfq  -%g°q' 
-^f"p'-y"p'    +iVr/ 
--hg' pq-%9' pq'  -^  ^g'piq  +  '^q') 

-  \^h°  q^  -  ^h°  q'^  +  \h°  {^  q"  -  4.qq'  +  iq'^) 

+  ^J'''q'  +  U°'q"  -9V/°'(2/+ii?^-8??'+8/^)). 


../ 


76  NOTES 

Hence  (6)  becomes,  on  substituting  <j,  a,  )8,  y  for  the  expressions  which  they  represent, 

[13]  C*  =  C-0-(-jLa  +  tL^+  iy  +  ^f'f 

Art.  26,  p.  41.      Derivation  of  formula  [14]. 

This  formula  is  derived  at  once  by  adding  formulae  [11],  [12],  [13].  But,  as 
Gauss  suggests,  it  may  also  be  derived  from  [6],  p.  38.  By  priming  the  q&  in  [6] 
we  obtain  a  series  for  (t/('+^').      Subtracting   this   series    from   [6],  and  noting  that 


r[/^   /  ^  —  <^'+ »/(  + TT  — t/('=j4 +5 +  C,  we  have,  correct  ^  terms  of  the  fifth  degree 

'''  \  (1)  A^B^C='n-iy{q-<^)(,r  +  |/>  +  i/V  +  I /i'l?  + /) 

'  -  |/°2 (^2  +  2  /  +  2  ^^'  +  2  q'"-)) 

The  second  term  on  the  right  in  (1)  may  be  written 

+  i«p(i-i/°  (?^'+  ?'+  ^/+ ^"))  •  2(-/°-f/>  -i/'y-i/i'(?'+  ?') 

-g°{q  +  q')-h°{q^+qq'+ q'^) 

+  y°\+q'  +  qq'  +  q'')'), 
of  which  the  last  factor  may  be  thrown  into  the  form : 
(-1/° 


-l/° 

-i/° 

-l/> 

-i/> 

-i9°q 

-f^°/ 

-\f"f 

-f/'y  +i/'y 

-%g'fq 

-i/p?'+i/p(^+/) 

-li-h°q- 

-i^A°/^+2/i°(^^  +  /^- 

-?/) 

^\rt 

+i/°^^"-i/°^(?^+?'^- 

-qq') 

Hence,  by  substituting  cr,  a,  /8,  y  for  the  expressions  they  represent,  (1)  becomes 
[14]  ^+5+C=7r  +  o-(|a+|y8+i7  +  i/'y 

Art.  27,  p.  42.      Omitting  terms  above  the  second  degree,  we  have 
a^  =  q'^—2qq'  +  q'^,       b'^  —  p^ -{■  q' ^ ,       &=f--Vq^. 

The    expressions  in  the  parentheses   of  the  first  set  of  fonnuliTe  for  A:'\  B*,  C* 
in  Art.  27  may  be  arranged  in  the  following  manner : 

{2j/-f       +4.qq'-     q")-i      [p"  +  q'')  +  {p' ^  q')  -2{q' -  2  qq' +  q")), 
(//    -2q^-^2qq'^     q'^)  -  (  2(/  +  q"^)  -  (p^  +  q")  -  (?"  -  2  qq'  +  /-)), 
if     ^  q^      ■\-  2qq'  -2q'^)=^\-{p'  +  q'^)  +  2{p''  +  q^)-{q^-2qq'  ^  q'^Y). 


>-^ 


NOTES  77 

Now    substituting   c^,  V^,  c^  for   {q^  —  2  qq'  +  q'^),  {p^+q''^),  ijP' +  q^)  respectively,  and 
changing  the  signs  of  both  members  of  the  last  two  of  these  equations,  we  have 

i^f-f  ■^^qq'-q"')  =  {b'' +  c^  -  2  a% 
-iy  -2q^+2qq'+  q'^)  ={a^  +  c^-2h''), 
-{f    ^  f     +2qq'-2q'^)={a^+h^-2c''). 

And   replacing   the   expressions   in   the   parentheses   in   the   first   set   of    formulae   for 
A*,  B*,  C*  by  their  equivalents,  we  get  the  second  set. 

Art.  27,  p.  42.     f°  =~  o^'  /"  ~  ^j  ^^^-j  "^^J  ^^  obtained  directly,  without  the 

use  of  the  general  considerations  of  Arts.  25  and  26,  in  the  following  way.      In  the  ^  S*-^ 

case  of  the  sphere 

rfs^=cos='(-|-)-rf/+ J/, 

hence 

«=cos(-|)  =l-^+2l^-etc., 

i.  e., 

f°^~2W        '^°^24^'       f'=9°=j"'=9'=^-         [Wangerin.] 

Art.  27,  p.  42,  1.  16.  This  theorem  of  Legendre  is  found  in  the  Memoires  (His- 
toire)  de  I'Academie  Royale  de  Paris,  1787,  p.  358,  and  also  in  his  Trigonometry, 
Appendix,  §  V.      He  states  it  as  follows  in  his  Trigonometry  : 

The  very  slightly  curved  spherical  triangle,  whose  angles  are  A,  B,  C  and  whose  sides 
are  «,  h,  c,  always  corresponds  to  a  rectilinear  triangle,  whose  sides  a,  h,  c  are  of  the  same 
lengths,  and  tvhose  opposite  angles  are  A  —  -^  e,  B — ^e,  C — -|-e,  e  being  the  excess  of  the 
sum  of  the  angles  in  the  given  spherical  triangle  over  two  right  angles. 

Art.  28,  p.  43,  1.  7.  The  sides  of  this  triangle  are  Hohehagen-Brocken,  Insel- 
berg-Hohehagen,  Brocken-Inselberg,  and  their  lengths  are  about  107,  85,  69  kilometers 
respectively,  according  to  Wangerin. 

Art.  29,  p.  43.      Derivation  of  the  relation  between  cr  and  o-*. 

In  Art.  28  we  found  the  relation 

A*^A--^a{2a+fi  +  y). 
Therefore 

sin  A*  =  sin  A  cos  (^  cr{2  a+/3  +  y))  —  cos  A  sin  (^  cr  (2  a  +  /8  +  y)), 

which,  after  expanding  cos  (^^  cr(2  a  +y8  +  y))  and  sin  (^  cr  (2  a  +/8  +  y))  and  reject- 
ing powers  of  C-Y^a-{2  a  +  /3  +  y))  above  the  first,  becomes 


-^  ^S^'^ 


78  NOTES 

(1)  sin  ^*  =  sin  4 -cos  4.  {■^a{2  a+ /3  + y)), 

correct  to  terms  of  the  fourth  degree. 
y-^J"     >v   l\^  V      '^,  But,  since  a  and  cr*  differ  only  by  terms  above  the  second  degree,  we  may  replace 

\  V\'^  ^^  ;A  '^  . Jn  (1)  cr  by  the  value  of  cr*,  ^  be  sin  A*.     We  thus  obtain,  with  equal  exactness, 

"^'^.(ijA-^   (2)  sin^  =  sin^*(l  + JjJccos7l.(2a+/3  +  7)). 

,   x'-N^?'       "■        Substituting  this  value  for  sin  J.  in  [9],  p.  40,  we  have,  correct  to  terms  of  the  sixth 

(  '"'V  degree,  the  first   formula  for  cr  given  in  Art.   29.     Since  2  5c  cos  .4*,  or  b^  +  c^  —  a^, 

V.  ."1  '^'    differs  from  2  5c  cos .4  only  by  terms  above  the  second  degree,  Ave  may  replace  2  5c  cosil 

C  1^^     '^i  in   this    formula   for   cr  by  P  +  c'^  —  a'^.      Also    cr*  =  ^  5c  sin  J^*.      Hence,  if  we    make 

these    substitutions    in    the   first   formula   for   cr,  we   obtain    the    second    formula  for  cr 

with   the   same   exactness.      In   the   case    of    a   sphere,   where    a=^  =  'y,   the   second 

formula  for  cr  reduces  to  the  third. 

When  the  surface  is  spherical,  (2)  becomes 

a, 

sin  A  =  sin^*  (1  +  ^  5c  cos  ^1). 

And  replacing  2  5ecosil  in  this  equation  by  (5^+c^— a^),  we  have 

sin  tI  =  sin  ^*  (1  +  ^  ( 5^  +  c'^  -  a")), 
or 

And  likewise  we  can  find 

S^  =  (l  +  E  (»"+<' -*')).       ^  =  (1  +  B(«'+ *'-''))• 

y     Multiplying  together  the   last  three    equations   and   rejecting  the    terms   containing  a^ 
and  a^,  we  have 

1  +  _^  (,,2  ,   .2  ,   ,^  _  sin^    .sin^   .sin  (7 
1  -1-  -^2  ^*  ^  ^  ^  ^  i  -  sin  A* .  sinB*  .  sin  C*' 

Finally,  taking  the  square  root  of  both  members  of  this   equation,  we  have,  with  the 
same  exactness, 

1  _L   ^  /  2  .    i2  J     2\         I  i^^^  ^   .  sin  j5   .  sin  C  V 
24  ^  \  ^sin  A* .  sm  B^ .  sin  C*' 

The   method    here    used    to   derive   the   last   formula   from    the    next   to   the   last 
formula  of  Art.  29  is  taken  from  Wangerin. 


NEUE 
ALLGEMEINE  UNTERSUCHUNGEN 

tJBER 

DIE  KRUMMEN   FLACHEN 
[1825] 

PUBLISHED   POSTHUMOUSLY   IN    GAUSS'S    WORKS,    VOL.    VIII,    1901.    PAGES   408-443 


NEW   GENERAL  INVESTIGATIONS 

OF 

CURVED  SURFACES 

[1825] 


Although  the  real  purpose  of  this  work  is  the  deduction  of  new  theorems  con- 
cerning its  subject,  nevertheless  we  shall  first  develop  what  is  already  known,  partly 
for  the  sake  of  consistency  and  completeness,  and  partly  because  our  method  of  treat- 
ment is  different  from  that  which  has  been  used  heretofore.  We  shall  even  begin  by 
advancing  certain  properties  concerning  plane  curves  from  the  same  principles. 

1. 

In  order  to  compare  in  a  convenient  manner  the  diflTerent  directions  of  straight 
lines  in  a  plane  with  each  other,  we  imagine  a  circle  with  unit  radius  described 
in  the  plane  about  an  arbitrary  centre.  The  position  of  the  radius  of  this  circle, 
drawn  parallel  to  a  straight  line  given  in  advance,  represents  then  the  position  of  that 
line.  And  the  angle  which  two  straight  lines  make  with  each  other  is  measured  by 
the  angle  between  the  two  radii  representing  them,  or  by  the  arc  included  between 
their  extremities.  Of  course,  where  precise  definition  is  necessary,  it  is  specified  at 
the  outset,  for  every  straight  Hne,  in  what  sense  it  is  regarded  as  drawn.  Without 
such  a  distinction  the  direction  of  a  straight  line  would  always  correspond  to  two 
opposite  radii. 


In  the  auxiliary  circle  we  take  an  arbitrary  radius  as  the  first,  or  its  terminal 
point  in  the  circumference  as  the  origin,  and  determine  the  positive  sense  of  measur- 
ing the  arcs  from  this  point  (whether  from  left  to  right  or  the  contrary)  ;  in  the 
opposite  direction  the  arcs  are  regarded  then  as  negative.  Thus  every  direction  of  a 
straight  line  is  expressed  in  degrees,  etc.,  or  also  by  a  number  which  expresses  them 
in  parts  of  the  radius. 


82  KARL  FREEDRICH  GAUSS 

Such  lines  as  diflPer  in  direction  by  360°,  or  by  a  multiple  of  360°,  have,  there- 
fore, precisely  the  same  direction,  and  may,  generally  speaking,  be  regarded  as  the 
same.  However,  in  such  cases  where  the  manner  of  describing  a  variable  angle  is 
taken  into  consideration,  it  may  be  necessary  to  distinguish  carefully  angles  differing 
by  360°. 

If,  for  example,  we  have  decided  to  measure  the  arcs  from  left  to  right,  and  if 
to  two  straight  lines  I,  V  correspond  the  two  directions  L,  L',  then  L'—  L  is  the  angle 
between  those  two  straight  lines.  And  it  is  easily  seen  that,  since  L'  —  L  falls 
between  —  180°  and  +  180°,  the  positive  or  negative  value  indicates  at  once  that  /' 
lies  on  the  right  or  the  left  of  I,  as  seen  from  the  point  of  intersection.  This  will 
be  determined  generally  by  the  sign  of  sin(X'— X). 

If  a  a'  is  a  part  of  a  curved  line,  and  if  to  the  tangents  at  a,  a'  correspond 
respectively  the  directions  a,  a',  by  which  letters  shall  be  denoted  also  the  corres- 
ponding points  on  the  auxiliary  circles,  and  if  A,  A'  be  their  distances  along  the  arc 
from  the  origin,  then  the  magnitude  of  the  arc  a  a'  or  A'  —  A  is  called  the  amplitude 
of  a  a'. 

The  comparison  of  the  amplitude  of  the  arc  a  a'  with  its  length  gives  us  the 
notion  of  curvature.  Let  I  be  any  point  on  the  arc  a  a',  and  let  \,  A  be  the  same 
with  reference  to  it  that  a,  A  and  a'.  A'  are  with  reference  to  a  and  a'.  If  now 
aX  or  A  —  A  be  proportional  to  the  part  al  of  the  arc,  then  we  shall  say  that  a  a'  is 
uniformly  curved  throughout  its  whole  length,  and  we  shall  call 

A-^ 
al 

the  measure  of  curvature,  or  simply  the  curvature.  We  easily  see  that  this  happens 
only  when  a  a'  is  actually  the  arc  of  a  circle,  and  that  then,  according  to  our  defini- 
tion, its  curvature  will  be   ±  -?  if  r   denotes  the   radius.      Since  we  always  regard   r 

as  positive,  the  upper  or  the  lower  sign  will  hold  according  as  the  centre  lies  to  the 
right  or  to  the  left  of  the  arc  ua'  (a  being  regarded  as  the  initial  point,  a'  as  the 
end  point,  and  the  directions  on  the  auxiliary  circle  being  measured  from  left  to 
right).  Changing  one  of  these  conditions  changes  the  sign,  changing  two  restores  it 
again. 

On  the  conti'ary,  if  A  —  A  be  not  proportional  to  a  I,  then  we  call  the  arc  non- 
uniformly  curved  and  the  quotient 

A-A 
al 


NEW  GENERAL  INVESTIGATIONS  OF  CURVED  SITRFACES  [1825]  83 

may  then  be  called  its  mean  curA'ature.  Curvature,  on  the  contrary,  always  presup- 
poses that  the  point  is  determined,  and  is  defined  as  the  mean  curvature  of  an  element 
at  this  point;  it  is  therefore  equal  to 

dK 

dal 

We  see,  therefore,  that  arc,  amplitude,  and  curvature  sustain  a  similar  relation  to  each 
other  as  time,  motion,  and  velocity,  or  as  volume,  mass,  and  density.  The  reciprocal 
of  the  curvature,  namely, 

dal 

is  called  the  radius  of  curvature  at  the  point  I.  And,  in  keeping  with  the  above 
conventions,  the  curve  at  this  point  is  called  concave  toward  the  right  and  convex 
toward  the  left,  if  the  value  of  the  curvature  or  of  the  radius  of  curvature  happens 
to  be  positive ;  but,  if  it  happens  to  be  negative,  the  contrary  is  true. 

3. 

If  we  refer  the  position  of  a  point  in  the  plane  to  two  perpendicular  axes  of 
coordinates  to  which  correspond  the  directions  0  and  90°,  in  such  a  manner  that  the 
first  coordinate  represents  the  distance  of  the  point  from  the  second  axis,  measured  in 
the  direction  of  the  first  axis  ;  whereas  the  second  coordinate  represents  the  distance 
from  the  first  axis,  measured  in  the  direction  of  the  second  axis ;  if,  further,  the  inde- 
terminates  x,  y  represent  the  coordinates  of  a  point  on  the  curved  line,  s  the  length 
of  the  line  measured  from  an  arbitrary  origin  to  this  point,  ^  the  direction  of  the 
tangent  at  this  point,  and  r  the  radius  of  curvature  ;    then  we  shaU  have 

Ja;  =  cos  (^  .  ds, 
dy  =  sin  ^  .  ds, 
ds 

If  the  nature  of  the  curved  line  is  defined  by  the  equation  T  =  0,  where  V  is  a 
function  of  x,  y,  and  if  we  set 

d  V  =  pdx  +  q  dy, 
then  on  the  curved  line 

pdx  -\-  qdy^=^  ^. 
Hence 

p  cos  (/>  +  J*  sin  ^  =  0, 


84  KARL  FRIEDRICH  GAUSS 

and  therefore 

P 
tan<A  =  — -• 

^  <1 

We  have  also 

cos  ^  .  dp  +  sin  <f)  .  dq  —  {p  sin  <^  —  §'  cos  <^)  </  ^  =  0. 
If,  therefore,  we  set,  according  to  a  well  known  theorem, 

dp  =P  dx  +  Q  dy, 

dq=  Qdx  +  R  dy, 
then  we  have 

(P  cos^  ^  +  2  ^  cos  <^  sin  <^  +  i?  sin'^  <^)  ds  =  {psin(fi  —  q  cos  ^)  c?  (j), 
therefore 

1  _  P  cos^  ^  +  2  ^  cos  <^  sin  ^  +E  sin^  ^ 
r  jo  sin  (^  —  §•  cos  <f)  ' 

or,  since 

^l_Pg^-2^jt?y+Jg/ 


The  ambiguous  sign  in  the  last  formula  might  at  first  seem  out  of  place,  but 
upon  closer  consideration  it  is  found  to  be  quite  in  order.  In  fact,  since  this  expres- 
sion depends  simply  upon  the  partial  differentials  of  V,  and  since  the  function  V  itself 
merely  defines  the  nature  of  the  curve  without  at  the  same  time  fixing  the  sense  in 
which  it  is  supposed  to  be  described,  the  question,  whether  the  curve  is  convex 
toward  the  right  or  left,  must  remain  undetermined  until  the  sense  is  determined  by 
some  other  means.  The  case  is  similar  in  the  determination  of  (j)  by  means  of  the 
tangent,  to  single  values  of  which  correspond  two  angles  differing  by  180°.  The 
sense  in  which  the  curve  is  described  can  be  specified  in  the  following  different  ways. 

I.  By  means  of  the  sign  of  the  change  in  x.  If  x  increases,  then  cos  (j>  must  be 
positive.  Hence  the  upper  signs  will  hold  if  q  has  a  negative  value,  and  the  lower 
signs  if  q  has  a  positive  value.      When  x  decreases,  the  contrary  is  true. 

II.  By  means  of  the  sign  of  the  change  in  y.  If  y  increases,  the  upper  signs 
must  be  taken  when  p  is  positive,  the  lower  when  p  is  negative.  The  contrary  is 
true  when  y  decreases. 

III.  By  means  of  the  sign  of  the  value  which  the  function  V  takes  for  points 
not  on  the  curve.      Let  8x,  8y  be  the  variations   of  x,  y  when  we  go  out  from  the 


NEW  GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  [1825]  85 

curve  toward  the  right,  at  right  angles  to  the  tangent,  that  is,  in  the  direction 
<f)  +  90°  ;  and  let  the  length  of  this  normal  be  8 p.      Then,  evidently,  we  have 

8x  =  Sp  .  cos(</)  +  90°), 
8^  =  8  p  .  sin  (i  +  90°), 
or 

Sx  =  —  8/3  .  sin <p, 

S?/  =  +  8 p  .  cos  (f). 

Since  now,  when  8  p  is  infinitely  small, 

8  F=  j9  8a;  +  §■  8y 

=  (— j»  sintf)  +  q  cos  <ji)  8  p  ',y 

=  ^8pV{f^t) 

and  since  on  the  curve  itself  V  vanishes,  the  upper  signs  will  hold  if  V,  on  passing 
through  the  curve  from  left  to  right,  changes  from  positive  to  negative,  and  the  con- 
trary. If  we  combine  this  with  what  is  said  at  the  end  of  Art.  2,  it  follows  that  the 
curve  is  always  convex  toward  that  side  on  which   V  receives  the  same  sign  as 

For  example,  if  the  curve  is  a  circle,  and  if  we  set 

r=  a:^  +  /  -  a^ 
then  we  have 

P  =  2,         $  =  0,         R  =  % 

(/  +  fY-=  8  a^ 
r  =  ±  « 

and  the  curve  wiU  be  convex  toward  that  side  for  which 

^^ + y  >  «^ 

as  it  should  be. 

The  side  toward  which  the  cui've  is  convex,  or,  what  is  the  same  thing,  the  signs 
in  the  above  formulae,  will  remain  unchanged  by  moving  along  the  curve,  so  long  as 

8V_ 
8p 

does  not  change  its  sign.  Since  F  is  a  continuous  function,  such  a  change  can  take 
place  only  when  this  ratio  passes  through  the  value  zero.  But  this  necessarily  pre- 
supposes that  p  and  q  become  zero  at  the  same  time.      At  such  a  point  the  radius 


86  KAEL  FRIEDEICH  GAUSS 

of  curvature   becomes  infinite   or   the  curvature   vanishes.      Then,  generally  speaking, 
since  here 

—  f  sin  <]>  +  q  cos  <f> 

wUl  change  its  sign,  we  have  here  a  point  of  inflexion. 


The  case  where  the  nature  of  the    curve  is    expressed  by    setting  ^  equal  to   a 
given  function  of  x,  namely,  i/—X,  is  included  in  the  foregoing,  if  we  set 

V=X-i/. 


If  we  put 
then  we  have 

therefore 


dX==X'dx,  dX'  =  X"dx, 

p=X',  ^  =  -1, 

P  =X",  Q  =  0,  i?  =  0, 

1  X" 


-r    {i+xy^ 

Since  q  is  negative  here,  the  upper  sign  holds  for  increasing  values  of  x.  We  can 
therefore  say,  briefly,  that  for  a  positive  X"  the  curve  is  concave  toward  the  same 
side  toward  which  the  ^-axis  lies  with  reference  to  the  a:-axis ;  while  for  a  negative 
X"  the  curve  is  convex  toward  this  side. 


If  we   regard   x,  y  as  functions   of  s,  these   formulae    become   stUl   more    elegant. 
Let  us  set 


Then  we  shall  have 


dx         , 

dx' 
ds 

=  x", 

ds  y ' 

dy' 
ds 

-y"- 

x'  =  COS  <^, 

y' 

—  sin  <^, 

r 

y" 

COS<^  _ 

r 

rx  ,  x=ry 


NEW  GENERAL  UsTVESTIGATIONS  OF  CURVED  SURFACES  [1825]  8 


7/ 


or  also 
so  that 
represents  the  curvature,  and 

1 

=  r{x'tj"-y' 
x'y"-y'a/' 
1 

the  radius  of  curvature. 

^y"-y'x" 

7. 

We  shall  now  proceed  to  the  consideration  of  curved  surfaces.  In  order  to  repre- 
sent the  directions  of  straight  lines  in  space  considered  in  its  three  dimensions,  we 
imagine  a  sphere  of  unit  radius  described  about  an  arbitrary  centre.  Accordingly,  a 
point  on  this  sphere  will  represent  the  direction  of  all  straight  lines  parallel  to  the 
radius  whose  extremity  is  at  this  point.  As  the  positions  of  all  points  in  space 
are  determined  by  the  perpendicular  distances  x,  y,  z  from  three  mutually  perpendicu- 
lar planes,  the  directions  of  the  three  principal  axes,  which  are  normal  to  these 
principal  planes,  shall  be  represented  on  the  auxiliary  sphere  by  the  three  points 
(1),  (2),  (3).  These  points  are,  therefore,  always  90°  apart,  and  at  once  indicate  the 
sense  in  which  the  coordinates  are  supposed  to  increase.  We  shall  here  state  several 
well  known  theorems,  of  which  constant  use  will  be  made. 

1)  The  angle  between  two  intersecting  straight  lines  is  measured  by  the  arc  [of 
the  great  circle]    between  the  points  on  the  sphere  which  represent  their   dii-ections. 

2)  The  orientation  of  every  plane  can  be  represented  on  the  sphere  by  means 
of  the  great  circle  in  which  the  sphere  is  cut  by  the  plane  through  the  centre  parallel 
to  the  first  plane. 

3)  The  angle  between  two  planes  is  equal  to  the  angle  between  the  great  cir- 
cles which  represent  their  orientations,  and  is  therefore  also  measured  by  the  angle 
between  the  poles  of  the  great  circles. 

4)  If  X,  y,  z ;  x',  y',  z'  are  the  coordinates  of  two  points,  r  the  distance  between 
them,  and  L  the  point  on  the  sphere  which  represents  the  direction  of  the  straight 
line  drawn  from  the  first  point  to  the  second,  then 

x'  =  X  +  r  cos(l)i/, 
y'  =  y  -\-  r  cos(2)j&, 
2'  =  2  +  r  cos(3)X. 

5)  It  follows  immediately  from  this  that  we  always  have 

cosXl)i  +  cos''(2)i:  +  cos^(3)Z  -  1 


+\ 


88  KARL  FRIEDRICH  GAUSS 

[and]   also,  if  L'  is  any  other  point  on  the  sphere, 

cos(l)X  .  cos(l)X'  +  cos(2)i:  .  cos(2)i'  +  cos(3)Z  .  cos(3)Z'  =  cosXZ'. 

We  shall  add  here  another  theorem,  which  has  appeared  nowhere  else,  as  far  as 
we  know,  and  which  can  often  be  used  with  advantage. 

Let  X,  L',  L",  L'"  be  fbur  points  on  the  sphere,  and  A  the  angle  which  LL'" 
and  L'  L"  make  at  their  point  of  intersection.     [Then  we  have] 

cosXi'  .  cos  L"L"'- COS  LL"  .  cos  L' L'"  =  sin  LL'"  .  sin  L'L"  .cos  A. 

The  proof  is  easily  obtained  in  the  following  way.     Let 

AL=t,  AL'^f,  AL"^t",  AL"'^t'"; 

we  have  then 

cos  LL'      =cost    cost'    +sin^    sinf    cos  vl, 
cos  L"L"'  =  cos  t"  cos  f"  +  sin  t"  sin  /'"  cos  A, 
cos  LL"     =cost    cost"  +sin^     sinf  cos  ^, 
\**  cos  L'L'"  =  cos  f  cos  f"  +  sin  f   sin  f"  cos  A. 

Therefore 

cos  LL'  cos  L"L'"  -  cos  LL"  cos  L'L" 

I  =  cos  A  \  cos  t  cos  f  sin  t"  sin  f"  +  cos  t"  cos  f"  sin  t  sin  t' 

'-^'  U*  —  COS  t  cos  t"  sin  t'  sin  f"  —  cos  f  cos  f"  sin  t  sin  ^"  | 

=  cos  A  (cos  ^  sin  t'"  —  cos  if'"  sin  t)  (cos  f  sin  t"  —  cos  i"  sin  t') 
=  cos  ^  sin  (t'"  —t )  sin  (if"  —  f) 
=  GosA  sin  LL'"  sin  L'L". 

Since  each  of  the  two  great  circles  goes  out  from  A  in  two  opposite  directions, 
two  supplementary  angles  are  formed  at  this  point.  But  it  is  seen  from  our  analysis 
that  those  branches  must  be  chosen,  which  go  in  the  same  sense  from  L  toward  L'" 
and  from  L'  toward  L". 

Instead  of  the  angle  A,  we  can  take  also  the  distance  of  the  pole  of  the  great 
circle  LL'"  from  the  pole  of  the  great  circle  L' L".  However,  since  every  great  circle 
has  two  poles,  we  see  that  we  must  join  those  about  which  the  great  circles  run  in 
the  same  sense  from  L  toward  L'"  and  from  L'  toward  L",  respectively. 

The  development  of  the  special  case,  where  one  or  both  of  the  arcs  L  L'"  and 
L'  L"  are  90°,  we  leave  to  the  reader. 

6)  Another  useful  theorem  is  obtained  from  the  following  analysis.  Let  L,  L', 
L"  be  three  points  upon  the  sphere  and  put 


L- 


NEW  GElSnERAL  INVESTIGATIONS  OF  CURVED  SURFACES  [1825J  89 

cos  L  (1)  =  X,  cos  L  (2)  =  y,  cos  L  (3)  =  s, 
cos  L'  (1)  =  x',  cos  L'  (2)  =  /,  cos  L'  (3)  =  z' , 
cos  X"(l)  =  x",     cos  X"  (2)  =  y",     cos  X"  (3)  =  z" . 

We  assume  that  the  points  are  so  arranged  that  they  run  around  the  triangle 
included  by  them  in  the  same  sense  as  the  points  (1),  (2),  (3).  Further,  let  \  be 
that  pole  of  the  great  circle  L'  L"  which  lies  on  the  same  side  as  L.  We  then  have, 
from  the  above  lemma, 

y'  z"  -  z'  y"  =  sin  L'  L"  .  cos  \(1), 

z'  x"  -  x'  z"  =  sin  L'  L"  .  cos  \(2), 

x'y"  -  y'  x"  =  sin  L'  L"  .  cos  X(3). 

Therefore,  if  we  multiply  these  equations  by  x,  y,  z  respectively,  and  add  the  pro- 
ducts, we  obtain 

xy'  z"  +  x' y" z  +  x" y  z'  —  xy"  z'  —  x' y  z"  —  x" y' z=  s'mL'  L" .  cos  \L, 

wherefore,  we  can  write  also,  according  to  weU  known  principles  of  spherical  trigo- 
nometry, _- 

sin  L'  L" .  sin  LI/^.shxL'  - 

=  sin  L'  L"  .  sin  L  L'  .  sin  L"  - 

=  sinX'Z".  sini/'X".  sinX,      v 

if  L,  L' ,  L"  denote  the  three  angles  of  the  spherical  triangle.  At  the  same  time  we 
easily  see  that  this  value  is  one-sixth  of  the  pyramid  whose  angular  points  are  the 
centre  of  the  sphere  and  the  three  points  L,  L',  L"  (and  indeed  positive,  if  etc.). 

The  nature  of  a  curved  surface  is  defined  by  an  equation  between  the  coordinates 
of  its  points,  which  we  represent  by 

/  {x,  y,  2)  =  0- 
Let  the  total  differential  of  /  {x,  y,  z)  be 

Pdx  +  Qdy  +  Rdz,        ;  ^'^^ 

where  P,  Q,  R  are  functions  of  x,  y,  z.  We  shall  always  distinguish  two  sides  of  the 
surface,  one  of  which  we  shall  call  the  upper,  and  the  other  the  lower.  Generally 
speaking,  on  passing  through  the  surface  the  value  of  /  changes  its  sign,  so  that,  as 
long  as  the  continuity  is  not  interrupted,  the  values  are  positive  on  one  side  and  nega- 
tive on  the  other. 


P'I'I 


I'l 


90  KAEL  FRIEDRICH  GAUSS 

The  direction  of  the  normal  to  the  surface  toward  that  side  which  we  regard  as 
the  upper  side  is  represented  upon  the  auxiliary  sphere  by  the  point  L.     Let 

cos  i;(l)  =X,  cos  L{2)  =  Y,  cos  X(3)  =  Z. 

Also  let  ds  denote  an  infinitely  small  line  upon  the  surface ;  and,  as  its  direction  is 
denoted  by  the  point  X  on  the  sphere,  let 

cos  X(l)  =  ^,  cos  \(2)  =  Tj,  cos  X(3)  =  t,. 

We  then  have  i 

dx  —  ^  ds,  di/  =  7)  ds,  ds  =  ^  ds,     ' " 

therefore      L;-\j_(j 

and,  since  XL  must  be  equal  to  90°,  we  have  also 

Since  P,  Q,  R,  X,  Y,  Z  depend  only  on  the  position  of  the  surface  on  which  we  take 
the  element,  and  since  these  equations  hold  for  every  direction  of  the  element  on  the 
surface,  it  is  easily  seen  that  P,  Q,  R  must  be  proportional  to  X,  Y,  Z.     Therefore 

P=X,i,  Q^Yfi,  R^Zfj,, 

Therefore,  since 

X'  +  Y^+Z'^l; 

^=PX+QY+RZ 
and 

fj?  =  P^  +  Q'' +  R", 
or 

IX  =^  ±i/{P'+Q'  +  R'). 

If  we  go  out  from  the  surface,  in  the  direction  of  the  normal,  a  distance  equal  to 
the  element  Sp,  then  we  shall  have 

Sx=X8p,  8f/  =  YSp,  Ss^ZSp 

and 

8/=P  Sz  +  QSt/  +RSs=  iihp. 

We  see,  therefore,  how  the  sign  of  /x  depends  on  the  change  of  sign  of  the  value  of 
/  in  passing  from  the  lower  to  the  upper  side. 

Let  us  cut  the  curved  surface  by  a  plane  through  the  point  to  which  our  nota- 
tion refers ;  then  we  obtain  a  plane  curve  of  which  ds  is  an  element,  in  connection 
with  which  we  shall  retain  the  above  notation.  We  shall  regard  as  the  upper  side  of 
the  plane  that  one  on  which  the  normal  to  the  curved  surface  lies.     Upon  this   plane 


NEW  GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  [1825]  91 

we  erect  a  normal  whose  direction  is  expressed  by  the  point  S  of  the  auxiliary 
sphere.  By  moving  along  the  curved  line,  X  and  L  wUl  therefore  change  their  posi- 
tions, whUe  S  remains  constant,  and  \L  and  X8  are  always  equal  to  90°.  Therefore 
\  describes  the  great  circle  one  of  whose  poles  is  S.     The  element  of  this  great  circle 

ds 
will  be  equal  to  — ,  if  r  denotes  the  radius   of  curvature  of  the  curve.      And  again, 

if  we  denote  the  direction  of  this  element  upon  the  sphere  by  X',  then  X'  wiU  evi- 
dently lie  in  the  same  great  circle  and  be  90°  from  X  as  well  as  from  S.  If  we 
now  set 

cos  X'(l)  -=  ^',  cos  X'(2)  -  -q',  cos  X'(3)  =  i', 

then  we  shall  have 

since,  in  fact,  f,  tj,  ^  are  merely  the  coordinates  of  the  point  X  referred  to  the  centre 
of  the  sphere. 

Since  by  the  solution  of  the  equation  f(x,  y,  z)^^^  the  coordinate  z  may  be 
expressed  in  the  form  of  a  function  of  x,  y,  we  shall,  for  greater  simplicity,  assume 
that  this  has  been  done  and  that  we  have  found 

z=F{x,y). 
We  can  then  write  as  the  equation  of  the  surface  Ki^ 

z-F{x,y)  =  %  r^J:-' 

or  ^^-       0^\ 

f{x,y,  2)  =2  -F{x, y).  .^J ■)  ' 

From  this  follows,  if  we  set  i  ..      v'^ 

dF{x,y)  =  tdx  +  u  dy, 
P=—t,  Q=^  —  u,  R  =  l, 

where  t,  u  are  merely  functions  of  x  and  y.     We  set  also 

dt=Tdx  +  Udy,  du=  Udx  +  Vdy. 

Therefore  upon  the  whole  surface  we  have 

dz  =  tdx  +  udy 
and  therefore,  on  the  curve,  ,  a  n 

C=t^+  urj.  ^■ 

Hence  differentiation  gives,  on  substituting  the  above  values  for  d^,  d-q,  dl„ 

{C'-te-U7j')y=idt  +  7)du 

=  {eT+2^7,U+r)'V)ds, 


^ 


I> 


92 
or 


KARL  FRIEDRICH  GAUSS 


1  _eT+2^r,U+'q^V 

_Z{eT+2ir,U+r,'V)_ 
COS  L  \' 


Aa^' 


10. 

Before  we  further  transform  the  expression  just  found,  we  will  make  a  few 
remarks  about  it. 

A  normal  to  a  curve  in  its  plane  corresponds  to  two  directions  upon  the  sphere, 
according  as  we  draw  it  on  the  one  or  the  other  side  of  the  curve.  The  one  direc- 
tion, toward  which  the  curve  is  concave,  is  denoted  by  X',  the  other  by  the  opposite 
point  on  the  sphere.  Both  these  points,  like  L  and  S,  are  90°  from  X,  and  there- 
fore lie  in  a  great  circle.  And  since  2  is  also  90°  from  X,  SX  =  90°  —  XX',  or 
=ZX'-90°.      Therefore 

cos  Lk'  =  ±  sin  SX, 

where  sin  S  Z  is  necessarily  positive.  Since  /  is  regarded  as  positive  in  our  analysis, 
the  sign  of  cos  L  X'  will  be  the  same  as  that  of 

And  therefore  a  positive  value  of  this  last  expression  means  that  L  X'  is  less  than 
90°,  or  that  the  curve  is  concave  toward  the  side  on  which  lies  the  projection  of  the 
normal  to  the  surface  upon  the  plane.  A  negative  value,  on  the  contrary,  shows  that 
the  curve  is  convex  toward  this  side.      Therefore,  in  general,  we  may  set  also 

l_Z{eT+2^7)U+7j^r) 
r  sin  SZ 

if  we  regard  the  radius  of  curvature  as  positive  in  the  first  case,  and  negative  in 
the  second.  SZ  is  here  the  angle  which  our  cutting  plane  makes  with  the  plane 
tangent  to  the  curved  surface,  and  we  see  that  in  the  different  cutting  planes  passed 
through  the  same  point  and  the  same  tangent  the  radii  of  curvature  are  proportional 
to  the  sine  of  the  inclination.  Because  of  this  simple  relation,  we  shall  limit  our- 
selves hereafter  to  the  case  where   this   angle  is  a  right  angle,  and  where  the  cutting 


NEW  GEKERAL  mVESTIGATIOlSrS  OF  CUEVED  STJREACES  [1825]  93 

Jiff 
plane,  therefore,  is  passed  through  the  normal  of  the  curved  surface.      Hence  we  have  ^  |      1  ^^ 

for  the  radius  of  curvature  the  simple  formula  -  •       -/■'"^^  , 

11. 


Os^ 


Since  an  infinite  number  of  planes  may  be  passed  through  this  normal,  it  follows 
that  there  may  be  infinitely  many  difi'erent  values  of  the  radius  of  curvature.  In  this 
case  T,  U,  V,  Z  are  regarded  as  constant,  f,  17,  I,  as  variable.  In  order  to  make  the 
latter  depend  upon  a  single  variable,  we  take  two  fixed  points  M,  M  90°  apart  on  the 
great  circle  whose  pole  is  L.  Let  their  coordinates  referred  to  the  centre  of  the  sphere 
be  a,  /8,  y  ;   a',  |S',  y.      We  have  then  ■  ^ -^  '  S"' 

cos  \(1)  =  cos  >:M.  cos  il!f(l)  +  cos  \M' .  cos  M'il)  +  cos  \L  .  cos  L{1):\  <V  ^ 
If  we  set 

XM=<f>, 


then  we  have 

and  the  formula  becomes 
and  likewise 

Therefore,  if  we  set 


cos  X  M'  =  sin  (f>, 

^  =  a  cos  (j)  +  a'  sin  (f>, 

7)  =/3  cos  (^  +  /S'  sin  <f), 
^  =  y  cos  <^  +  y'  sin  (ft. 

B^{aa'T+  {a'/3+  al3')U  + fi/3'V)Z, 

C  =  la'^T+2a'l3'U  +  l3"V)Z, 


we  shall  have 

' 

-  =il  Gos'^ff)  -\r  2  B  cos  (^  sin  (^  +  Csin^<^ 

J.  +  C     A  —  C 
—  — 2 ' 0 —  cos  2  (^  +  5  sin  2  <^. 

If  we  put 

^ (J 

— 2—  =  ^  cos  2  e, 

B  =  Bsm  2  9, 

94  KARL  FRIEDRICH  GAUSS 

A  —  C 
where  we  may  assume  that  U  has  the  same  sign  as  — « — >  then  we  have 

-  =  ^{A  +  C)  +1:  cos2{<f>-  d). 

It  is   evident   that  <^  denotes  the  angle  between  the  cutting  plane  and  another  plane 
through  this  normal  and  that  tangent  which  corresponds  to  the  direction  M.     Evidently, 

therefore,  -  takes  its  greatest  (absolute)  value,   or  r  its   smallest,  when  ^^^6;  and  - 

its  smallest  absolute  value,  when  (f)  —  0  +  90°.      Therefore  the  greatest  and  the   least 
curvatures    occur   in   two   planes   perpendicular   to  each    other.      Hence  these  extreme 

values  for  -  are 
r 


-^  ^(A+C)±^{(^)\b') 


Their  sum  is  .4  +  C  and  their  product  \8  AC  —  B^,  or  the  product  of  the  two  extreme 
radii  of  curvature  is 

_        1 

'~  AC-B"' 

This   product,   which    is    of   great    importance,   merits    a    more   rigorous    development.^ 
In  fact,  from  formulae  above  we  find 

AC-B'={a^  -^a:)^{TV-TP)Z\ 

But  from  the  third  formula  in   [Theorem]   6,  Art.  7,  we  easily  infer  that 

a^'-fia'^  ±Z, 


b.\lV         therefore 

Besides,  from  Art.  8 

therefore 


AC-B^  =  Z'{TV-IP). 
^  R 

_  ^  1 

TV-U' 


AC-B^  = 


Just  as  to  each  point  on  the  curved  surface  corresponds  a  particular  point   L  on 
the  auxiliary  sphere,  by   means  of  the   normal  erected  at  this  point  and  the  radius  of 


NEW  GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  [1825]  95 

the  auxiliary  sphere  parallel  to  the  normal,  so  the  aggregate  of  the  points  on  the 
auxiliary  sphere,  which  correspond  to  all  the  points  of  a  line  on  the  curved  surface, 
forms  a  line  which  will  correspond  to  the  line  on  the  curA^ed  surface.  And,  likewise, 
to  every  finite  figure  on  the  curved  surface  will  correspond  a  finite  figure  on  the 
auxiliary  sphere,  the  area  of  which  upon  the  latter  shall  be  regarded  as  the  measure 
of  the  amplitude  of  the  former.  We  shall  either  regard  this  area  as  a  number,  in 
which  case  the  square  of  the  radius  of  the  auxiliary  sphere  is  the  unit,  or  else 
express  it  in  degrees,  etc.,  setting  the  area  of  the  hemisphere  equal  to  360°. 

The  comparison  of  the  area  upon  the  curved  surface  with  the  corresponding 
amplitude  leads  to  the  idea  of  what  we  call  the  measure  of  curvature  of  the  sur- 
face. If  the  former  is  proportional  to  the  latter,  the  curvature  is  called  uniform ; 
and  the  quotient,  when  we  divide  the  amplitude  by  the  surface,  is  called  the  measure 
of  curvature.  This  is  the  case  when  the  curved  surface  is  a  sphere,  and  the  measure 
of  curvature  is  then  a  fraction  whose  numerator  is  unity  and  whose  denominator  is 
the  square  of  the  radius. 

We  shall  regard  the  measure  of  curvature  as  positive,  if  the  boundaries  of  the 
figures  upon  the  curved  surface  and  upon  the  auxiliary  sphere  run  in  the  same  sense ; 
as  negative,  if  the  boundaries  enclose  the  figures  in  contrary  senses.  If  they  are  not 
proportional,  the  surface  is  non-uniformily  curved.  And  at  each  point  there  exists  a 
particular  measure  of  curvature,  which  is  obtained  from  the  comparison  of  correspond- 
ing infinitesimal  parts  upon  the  curved  surface  and  the  auxiliary  sphere.  Let  dcr  be 
a  surface  element  on  the  former,  and  dt  the  corresponding  element  upon  the  auxiliary 
sphere,  then 

dt 
d(T 

will  be  the  measure  of  curvature  at  this  point. 

In  order  to  determine  their  boundaries,  we  first  project  both  upon  the  a;^-plane. 
The  magnitudes  of  these  projections  are  Z dcr,  Z d't.     The  sign  of  Z  wiU  show  whether  y^ 

the  boundaries  run  in  the  same  sense  or  in  contrary  senses  around  the  surfaces  and 
their  projections.  We  will  suppose  that  the  figure  is  a  triangle ;  the  projection  upon 
the  a;y-plane  has  the  coordinates 

x,y  ;  x-\-  dx,  y  -\-  dy  ;  x-r'hx,  y-\-  Sy. 

Hence  its  double  area  will  be 

2  Zda  ==dx  .8y  —  dy  .  8x. 
To  the  projection  of  the  corresponding  element  upon  the  sphere  will  correspond  the 
coordinates : 


96  KARL  FRIEDRICH  GAUSS 

X,  Y, 

dX  dX  BY  dY    , 

X+^-dx^-j^-di,,         Y-^^-dx  +  ^-dy, 

dX  dX  XT,     9^   s     4-  ^^    S 

From  this  the  double  area  of  the  element  is  found  to  be 

/ax    ,    ,  ax    ,  \/ar  ^       ar  ^  v 

The  measure  of  curvature  is,  therefore, 

dX    dY      dX    BY 


Since 


we  have 


dx      By         By      Bx 
X  =  -tZ,  Y=-uZ, 


^  dX^-Z\l+u^)dt+ZHu.du, 

p^,     ^(^A  -^  dY  =  +  Z^tu.dt-Z^{l  +  f)du, 

therefore 

^=Z^-{l  +  u^)T+tuU\,  ^==Z^tuT-il+f)U}, 

"^^Z^l-il  +  u^W+tuVl,  jy=Z^tuU-{l  +  f)V\, 

and 

w  ^Z'{TV-  U')  ((1  +  f)  (1  +  u')  -  fu') 
'    =Z'{TV-U^){l  +  f+u') 

^Z'{TV-U^) 

_     TV-IP 

"(l  +  f+u^f 

the  very  same  expression  which  we  have  found  at  the  end  of  the  preceding  article. 
Therefore  we  see  that 


KEW  GENERAL  INV:ESTIGATI0K'S  OF  CUEVED  SUEFACES  [1825]  97 

"  The  measure  of  curvature  is  always  expressed  by  means  of  a  fraction  whose 
numerator  is  unity  and  whose  denominator  is  -the  product  of  the  maximum 
and  minimum  radii  of  curvature  in  the  planes  passing  through  the  normal." 

12. 

We  will  now  investigate  the  nature  of  shortest  lines  upon  curved  surfaces.  The 
nature  of  a  curved  line  in  space  is  determined,  in  general,  in  such  a  way  that  the 
coordinates  x,  y,  z  of  each  point  are  regarded  as  functions  of  a  single  variable,  which 
we  shall  call  w.  The  length  of  the  curve,  measured  from  an  arbitrary  origin  to  this 
point,  is  then  equal  to 

If  we  allow  the  curve  to  change  its  position  by  an  infinitely  small  variation,  the  varia- 
tion of  the  whole  length  will  then  be 


/ 


dw  aw        "^       aw  aw  aw      ^       aw 


4mr^m-&  Mm-m-& 


dx  dy_ 


\  \  (t^)  +  ( jf )  +  (^)  }  \  i  W  +  \dJ  "^  ^dJ 


+  Is.d 


dz 

dw 


I   f  (dx\\  ldl\\  ldz_\' 
\  \\dw)  "^  \dwf  ^  \dwf 


The  expression  under  the  integral  sign  must  vanish  in  the  case  of  a  minimum,  as  we 
know.      Since  the  curved  line  lies  upon  a  given  curved  surface  whose  equation  is 

Pdx  +  Qdij^Rdz^Q, 

the  equation  between  the  variations  hx,  8//,  Ss 

PSz  +  Q8y  +  R8z^0 
must  also  hold.      From  this,  by  means  of  well  known  principles,  we  easily  conclude 
that  the  differentials 


98  KARL  FRIEDEICH  GAUSS 

dx  dy 

/I  div  ^  dw 


I    {  ldx\^      ((^M\\    /^^\M  I    (/^^\\    C^^V  ,    lds\^ 


rf. 


dtv 


V  {  (t?^)  +  {-£)  +  (i^) } 


must  be  proportional  to  the  quantities  P,  Q,  R  respectively.  If  ds  is  an  element  of 
the  curve ;  X  the  point  upon  the  auxiliary  sphere,  which  represents  the  direction  of 
this  element ;  L  the  point  giving  the  direction  of  the  normal  as  aboA'e ;  and  ^,17,  ^ ; 
X,  Y,  Z  the  coordinates  of  the  points  X,  L  referred  to  the  centre  of  the  auxiliary 
sphere,  then  we  have 

dx  =  ^  ds,         dp  =^r)  ds,         ds  ^=  t,  ds, 

Therefore  we  see  that  the  above  differentials  will  be  equal  to  d^,  drj,  dt,.  And  since 
P,  Q,  R  are  proportional  to  the  quantities  X,  Y,  Z,  the  character  of  the  shortest  line 
is  such  that 

d^ dfj dt, 

x~Y^'z' 

13. 

To  every  point  of  a  curved  line  upon  a  curved  surface  there  correspond  two 
points  on  the  sphere,  according  to  our  point  of  view ;  namely,  the  point  \,  which 
represents  the  direction  of  the  linear  element,  and  the  point  L,  which  represents  the 
direction  of  the  normal  to  the  surface.  The  two  are  evidently  90°  apart.  In  our 
former  investigation  (Art.  9),  where  [we]  supposed  the  curved  line  to  lie  in  a  plane, 
we  had  two  other  points  upon  the  sphere ;  namely,  S,  which  represents  the  direction 
of  the  normal  to  the  plane,  and  \',  which  represents  the  direction  of  the  normal  to 
the  element  of  the  curve  in  the  plane.  In  this  case,  therefore,  S  was  a  fixed  point 
and  \,  X'  were  always  in  a  great  circle  whose  pole  was  S.  In  generalizing  these 
considerations,  we  shall  retain  the  notation  S,  X',  but  we  must  define  the  meaning  of 
these  symbols  from  a  more  general  point  of  view.  When  the  curve  5  is  described, 
the  points  L,  X  also  describe  curved  lines  upon  the  auxiliary  sphere,  which,  gener- 
ally speaking,  are  no  longer  great  circles.      Parallel  to  the  element  of  the  second  line, 


NEW  GENERAL  INVESTIGATIONS  OF  CURVED  SURFACES  [1826]  99 

we  draw  a  radius  of  the  auxiliary  sphere  to  the  point  \',  but  instead  of  this  point 
we  take  the  point  opposite  when  \'  is  more  than  90°  from  Z.  In  the  first  case,  we 
regard  the  element  at  X  as  positive,  and  in  the  other  as  negative.  Finally,  let  S  be 
the  point  on  the  auxiliary  sphere,  which  is  90°  from  both  k  and  X',  and  which  is  so 
taken  that  X,  X',  S  lie  in  the  same  order  as  (1),  (2),  (3). 

The  coordinates  of  the  four  points  of  the  auxiliary  sphere,  referred  to  its  centre, 
are  for 


L 

X      Y 

z 

X 

i    V 

I 

X' 

e   v' 

u 

s 

a       13 

7- 

Hence  each  of  these  4 

points 

describes  a  ': 

line  upon 

the 

auxiliary  sphere, 

whose  elements 

we  shall 

express 

by 

dL, 

dX 

,  dX',  d 

s. 

We  have. 

,  therefore, 

d^ 

=-^'dX, 

dr) 

=  -q'dX, 

di 

=  i'dx. 

In  an  analogous  way  we  now  call 

dX 
ds 

the  measure  of  curvature  of  the  curved  line  upon  the  curved  surface,  and  its  reciprocal 

ds 

Hx 

the  radius  of  curvature.      If  we  denote  the  latter  by  p,  then 

pd^=^'dsj 
pd-q  =  7]'  ds, 
pdC  =  Cds. 

If,  therefore,  our  line   be  a   shortest  line,  f,  17',  £,'  must  be  proportional   to   the 
quantities  X,  Y,  Z.      But,  since  at  the  same  time 

we  have 

^'  =  ±X,    -n'^^Y,    C'-±z, 

and  since,  further, 

^'X+-q'Y+  C'Z=cosX'L 

^±{X'  +  Y'+Z') 
=  ±1, 


100  KARL  FRIEDRICH  GAUSS 

and  since  we  always  choose  the  point  X'  so  that 

X'X<90°, 
then  for  the  shortest  line 

or  \'  and  L  must  coincide.      Therefore 

pdr)^=Y  ds, 
pdt,  =^  Z  ds, 

V  and  we  have  here,  instead  of  4  curved  lines  upon  the  auxiliary  sphere^  only  3  to  con- 

sider. Every  element  of  the  second  line  is  therefore  to  be  regarded  as  lying  in  the 
great  circle  L\.  And  the  positiA^e  or  negative  value  of  p  refers  to  the  concavity 
or  the  convexity  of  the  curve  in  the  direction  of  the  normal. 

14. 

We  shall  now  investigate  the  spherical  angle  upon  the  auxiliary  sphere,  which 
the  great  circle  going  from  L  toward  \  makes  with  that  one  going  from  L  toward 
one  of  the  fixed  points  (1),  (2),  (3) ;  e.  g.,  toward  (3).  In  order  to  have  something 
definite  here,  we  shall  consider  the  sense  from  Z(3)  to  ZX  the  same  as  that  in  which 
(1),  (2),  and  (3)  lie.  If  we  call  this  angle  ^,  then  it  follows  from  the  theorem  of  Art. 
7  that 
-,  sinX(3)  .  sinZX  .  sin  «^=  Z^ — X-q, 

or,  since  X  X  =  90°  and 

&mL{?,)  =  V{X'  +  Y^)=^V{l-Z^), 


we  have 


Furthermore, 


or 


and 


.    ■  _    Y^-x-n 

*^      V{X^  +  Y^)' 

sin  X(3)  .  sin  ZX  .  cos  (^  =  ^, 

■    _  g 

^^'i'-  v{X^  +  Y^) 

Yi-Xri       I'       , 
tan  9  — j =  y-- 


4/ 


NEW  GENERAL  INVESTiaATIONS  OF  OUEVED  SURFACES  [1825]  101 

Hence  we  have 

^  {Yi-Xrjf+C 

The  denominator  of  this  expression  is 

=  r^r-  2  XY^T]  +X^yf  +  c^ 

=-  z^c' + (1  -z'')  (1  -  r)  +  r 
= 1  -  ^^ 

or 

^ ,  _iYd^-iXdyj  +  {X'n-Y^)di--ntdX+adY 

d<p— l—Z"" ■ 

"We  verify  readily  by  expansion  the  identical  equation 

■qt{X^  +  Y^+  Z^)  +  YZ{^^  +  r,^  +  C) 
=  {X^+Yyj+Zl){Zr}  +  YC)  +  {Xi-Z^){Xr]-Yi) 
and  likewise 

U{X^  +  Y^+Z^)  +XZ{e  +  rf^  +  C') 
^{X^  +  Yy,+ZQ{XC+Z^)  +  {Yi-Xr,){YC-Zr,). 
We  have,  therefore, 

y)C=-rz+{xc~z^){Xrj-Ya 

H=-XZ+{Y^-Xr^){YC-Zr)).       ■ 
Substituting  these  values,  we  obtain 

"9      I  —  z^^  YdX  —  XdY)  H -^  _^2 

+  ^^-^^  \di-  {Xt-  Z^)dX-  {Yl^- Zri)dY\. 

Now 

XdX+YdY^-ZdZ-=i), 

^dX  +7]dY  +  CdZ  -^-Xd^-Yd-q-ZdC. 

On  substituting  we  obtain,  instead  of  what  stands  in  the  parenthesis, 

dC-Z{Xd^+Ydri+ZdO. 
Hence 

di>  =  Y^^{YdX-XdY)  +  ^_^,  \CY-'qX^Z+  iXYZ\ 
-j^lCX+rtXYZ-^Y'Z} 
+  dUy^X-iY). 


?v 


,,x.-P 


102 

Since,  further. 


KARL  FRIEDRICH  GAUSS 


=  7)Z{i-z')  +  cyz\ 

rjXYZ-iY'Z  =  -^X^Z-i:XZ'-^Y^Z 

=^-^Z{l~Z^)-iXZ\ 
our  whole  expression  becomes 

d^  =  i-z^  {YdX-XdY) 

+  iXY-'nZ)d^  +  {^Z-iX)dri  +  {'qX~^Y)dl 

15. 

The  formula  just  found  is  true  in  general,  whatever  be  the  nature  of  the  curve. 
But  if  this  be  a  shortest  line,  then  it  is  clear  that  the  last  three  terms  destroy  each 
other,  and  consequently 

dj>==-  ^  5^,  {XdY-  YdX). 

But  we  see  at  once  that 

^^^^{XdY-YdX) 

is  nothing  but  the  area  of  the  part  of  the  auxiliary  sphere,  which  is  formed  between 
the  element  of  the  line  L,  the  two  great  circles   drawn   through  its  extremities  and 


(Z) 


W 


(J)    (Z) 


PP' 

(5) 


(J)    (2) 


(S) 


PP' 


(7) 


(3),  and   the    element  thus  intercepted  on  the  great  circle  through  (1)  and  (2).      This 
surface  is  considered  positive,  if  L  and  (3)  lie  on  the  same  side  of  (1)  (2),  and  if  the 


NEW  GENERAL  INTESTIGATIOiSrS  OF  CURAT^D  SURFACES  [1826]         103 


direction  from  P  to  P'  is  the  same  as  that  from  (2)  to  (1) ;  negative,  if  the  contrary 
of  one  of  these  conditions  hold ;  positive  again,  if  the  contrary  of  both  conditions  be 
true.  In  other  words,  the  surface  is  considered  positive  if  we  go  around  the  circum- 
ference of  the  figure  LL'P'P  in  the  same  sense  as  (1)  (2)  (3);  negative,  if  we  go 
in  the  contrary  sense. 

If  we  consider  now  a  finite  part  of  the  line  from  L  to  L'  and  denote  by  <^,  <^' 
the  values  of  the  angles  at  the  two  extremities,  then  we  have 

<^' =  </)  + Area  iX'P'P, 

the  sign  of  the  area  being  taken  as  explained. 

Now  let  us  assume  further  that,  from  the  origin  upon  the  curved  surface,  infinitely 
many  other  shortest  lines  go  out,  and  denote  by  A  that  indefinite  angle  which  the 
first  element,  moving  counter-clockwise,  makes  with  the  first  element  of  the  first  line ; 
and  through  the  other  extremities  of  the  different  curved  lines  let  a  curved  line  be  drawn, 
concerning  which,  first  of  all,  we  leave  it  undecided  whether  it  be  a  shortest  line  or 
not.  If  we  suppose  also  that  those  indefinite  values,  which 
for  the  first  line  were  (^,  <^' ,  be  denoted  by  \/»,  i/;'  for  each  of 
these  lines,  then  i/*'  —  i//  is  capable  of  being  represented  in 
the  same  manner  on  the  auxiliary  sphere  by  the  space 
LL\P\P.     Since  evidently  \\i  =  <l>—A,  the  space 

LL\P\P'L'L  =  x},'-y},-(}y'+  <!> 
=  xj,'  -<j)'+A 
=  LL\L'L+L'L\P\P'. 

If  the  bounding  line  is  also  a  shortest  line,  and,  when  prolonged,  makes  with 
LL',LL\  the  angles  B,B^;  if,  further,  x?  Xi  denote  the  same  at  the  points  L',L\, 
that  (^  did  at  L  in  the  line  LL',  then  we  have 

X,- x  + Area  Z' i:\P\P', 
y\,'-4>'^A=LL\L'L  +  x-X; 


p'p: 


but 


therefore 

B^-B^A=LL\L'L. 

The  angles  of  the  triangle  LL' L\  evidently  are 

A,  180°  -B,  B^ 


r 


f 


104  KARL  FRIEDRICH  GAUSS 

therefore  their  sum  is 

180°  +LL\L'L. 

The  form  of  the  proof  will  require  some  modification  and  explanation,  if  the  point 
(3)  falls  within  the  triangle.      But,  in  general,  we  conclude 

"  The  sum  of  the  three  angles  of  a  triangle,  which  is  formed  of  shortest  lines 
upon  an  arbitrary  curved  surface,  is  equal  to  the  sum  of  180°  and  the  area  of 
the  triangle  upon  the  anxiliary  sphere,  the  boundary  of  which  is  formed  by  the 
points  Z,  corresponding  to  the  points  in  the  boundary  of  the  original  triangle, 
and  in  such  a  manner  that  the  area  of  the  triangle  may  be  regarded  as  positive 
or  negative  according  as  it  is  inclosed  by  its  boundary  in  the  same  sense  as 
,  I  /  V  the  original  figure  or  the  contrary." 

Wherefore  we  easily  conclude  also  that  the  sum  of  all  the  angles  of  a  polygon 
of  n  sides,  which  are  shortest  lines  upon  the  curved  surface,  is  [equal  to]  the  sum 
of  (w  — 2)  180°  +  the  area  of  the  polygon  upon  the  sphere  etc. 

16. 

If  one  curved  surface  can  be  completely  developed  upon  another  surface,  then  all 
lines  upon  the  first  surface  will  evidently  retain  their  magnitudes  after  the  develop- 
ment upon  the  other  surface ;  likewise  the  angles  which  are  formed  by  the  intersec- 
tion of  two  lines.  Evidently,  therefore,  such  lines  also  as  are  shortest  lines  upon 
one  surface  remain  shortest  lines  after  the  development.  Whence,  if  to  any  arbi- 
trary polygon  formed  of  shortest  lines,  while  it  is  upon  the  first  surface,  there  cor- 
responds the  figure  of  the  zeniths  upon  the  auxiliary  sphere,  the  area  of  which  is 
A,  and  if,  on  the  other  hand,  there  corresponds  to  the  same  polygon,  after  its  devel- 
opment upon  another  surface,  a  figure  of  the  zeniths  upon  the  auxiliary  sphere,  the 
area  of  which  is  A',  it  foUows  at  once  that  in  every  case 

A^A'. 

Although  this  proof  originally  presupposes  the  boundaries  of  the  figures  to  be  short- 
est lines,  still  it  is  easily  seen  that  it  holds  generally,  whatever  the  boundary  may  be. 
For,  in  fact,  if  the  theorem  is  independent  of  the  number  of  sides,  nothing  wiU  pre- 
vent us  from  imagining  for  every  polygon,  of  which  some  or  all  of  its  sides  are  not 
shortest  lines,  another  of  infinitely  many  sides  all  of  which  are  shortest  lines. 

Further,  it  is  clear  that  every  figure  retains  also  its  area  after  the  transformation 
by  development. 


NEW  GEISTERAL  ESTVESTIGATIONS  OF  CURVED  SURFACES  [1825J  105 

We  shall  here  consider  4  figures  : 

1)  an  arbitrary  figure  upon  the  first  surface, 

2)  the  figure  on  the  auxiliary  sphere,  which  corresponds  to  the  zeniths  of  the 
previous  figure, 

3)  the  figure  upon  the  second  surface,  which  No.  1  forms  by  the  development, 

4)  the  figure  upon  the   auxiliary  sphere,  which   corresponds  to   the  zeniths  of 
No.  3. 

Therefore,  according  to  what  we  have  proved,  2  and  4  have  equal  areas,  as  also 
1  and  3.  Since  we  assume  these  figures  infinitely  small,  the  quotient  obtained  by 
dividing  2  by  1  is  the  measure  of  curvature  of  the  first  curved  surface  at  this  point, 
and  likewise  the  quotient  obtained  by  dividing  4  by  3,  that  of  the  second  surface. 
From  this  follows  the  important  theorem : 

"  In  the    transformation    of  surfaces    by  development  the  measure  of  curvature 
at  every  point  remains  unchanged." 
This  is  true,  therefore,  of  the  product  of  the  greatest  and  smallest  radii  of  curvature. 
In  the  case  of  the  plane,  the  measure  of  curvature  is  evidently  everywhere  zero. 
Whence  follows  therefore  the  important  theorem : 

"For  all  surfaces    developable   upon   a   plane  the  measure   of  curvature  every- 
where vanishes," 
or 

which  criterion  is  elsewhere  derived  from  other  principles,  though,  as  it  seems  to  us, 
not  with  the  desired  rigor.  It  is  clear  that  in  aU  such  surfaces  the  zeniths  of  aU 
points  can  not  fiU  out  any  space,  and  therefore  they  must  all  lie  in  a  Une. 

17. 

From  a  given  point  on  a  curved  surface  we  shaU  let  an  infinite  number  of  shortest 
lines  go  out,  which  shall  be  distinguished  from  one  another  by  the  angle  which  their 
first  elements  make  with  the  first  element  of  a  definite  shortest  line.  This  angle  we 
shall  call  9.  Further,  let  s  be  the  length  [measured  from  the  given  point]  of  a  part 
of  such  a  shortest  line,  and  let  its  extremity  have  the  coordinates  x,  y,  z.  Since  Q 
and  s,  therefore,  belong  to  a  perfectly  definite  point  on  the  curved  surface,  we  can 
regard  x,  y,  z  as  functions  of  B  and  s.  The  direction  of  the  element  of  s  corresponds 
to  the  point  \  on  the  sphere,  whose  coordinates  are  f,  f],  I,.      Thus  we  shall  have 


106  KARL  FRIEDRICH  GAUSS 

^~Vs'  "i-ds'  ^=d^- 

The  extremities  of  all  shortest  lines  of  equal  lengths  s  correspond  to  a  curved 
line  whose  length  we  may  call  t.  We  can  evidently  consider  t  as  a  function  of  s  and 
6,  and  if  the  direction  of  the  element  of  t  corresponds  upon  the  sphere  to  the  point  X' 
whose  coordinates  are  ^',  y\' ,  ^',  we  shall  have 

Consequently 

This  magnitude  we  shall  denote  by  u,  which  itself,  therefore,  will  be  a  function  of  9  and  s. 
We  find,  then,  if  we  differentiate  with  respect  to  s, 

du_^    dx        ^^y     dy       ^    dz_  \\dsf        \ds'        \dsf 


because 


©^©■-(ir-i. 


and  therefore  its  differential  is  equal  to  zero. 

But  since  all  points  [belonging]  to  one  constant  value  of  ^  lie  on  a  shortest  line, 
if  we  denote  by  L  the  zenith  of  the  point  to  which  s,  6  correspond  and  by  X,  Y,  Z 
the  coordinates  of  L,    [from  the  last  formulae  of  Art.  13], 

a^^x         ^^I_  d^s  ^z 

di~  f'  di  "  p'  ds^~  p' 

if  ]}  is  the  radius  of  curvature.      We  have,  therefore, 

But 

Xf +  FV  +  ^r  =  cosZ\'  =  0, 

because,  evidently,  X'  lies  on  the  great  circle  whose  pole  is  L.      Therefore  we  have 

as       ' 


NEW  G-El^ERAX  INVESTIGATIONS  OF  CURVED  SURFACES  [1825]         107 
or  u  independent  of  s,  and  therefore  a  function  of  6  alone.      But  for  s  =  0,  it  is   evi- 

7)  i 

dent  that ;!  =  0,    z-^  =  0,  and   therefore  z<  =  0.      Whence  we  conclude  that,  in  general, 

M  =  0,  or 

cos  XX'  =  0. 

From  this  follows  the  beautiful  theorem : 

"  If  all  lines   drawn  from  a  point  on  the   curved  surface  are  shortest  lines   of 
equal  lengths,  they  meet  the  line   which  joins  their  extremities   everywhere  at 
right  angles." 
We   can  show  in   a  similar  manner  that,  if  upon  the  curved  surface  any  curved 
line  whatever  is  given,  and  if  we  suppose  drawn  from  every  point  of  this  line  toward 
the  same  side  of  it  and  at  right  angles  to  it  only  shortest  lines   of  equal  lengths,  the 
extremities   of  which  are  joined  by  a  line,  this  line    will   be    cut  at   right  angles  by 
those  lines  in  aU  its  points.      We  need  only  let  Q  in  the  above  development  represent 
the  length  of  the  given  curved  line  from  an  arbitrary  point,  and  then  the  above  calcu- 
lations  retain   their   validity,  except   that  m  =  0   for   s  =  0    is    now    contained   in    the 
hypothesis. 

18. 

The  relations   arising  from  these  constructions   deserve  to  be  developed  still  more 

fully.      We  have,  in  the  first  place,  if,  for  brevity,  we  write  m  for  ^-g? 

(1) 

(2) 

(3) 

(4) 

(5) 

Furthermore, 

(6)  X''  +  r^   +Z''  =  1, 


ds ' 

^'          ds      '''         ds 

=  ^, 

dx 

w 

e    +V'    +C    =1, 

=  m 

and 


(7)  Xi  +Yr,  +ZC  =0, 

(8)  Xi'+Yri'  +  ZC'==0, 

X=Cv'-vC', 

[9]  i       Y^H'-a', 

z  =  'ne-H; 


108  KARL  FRIEDRICH  GAUSS 

r      ^'  =  r,Z-iY, 
[10]  {      'q'=iX-^Z, 


[11]  \    ■n  =  z^'-xv, 

{    i^x-n'-Y^'. 

Likewise,  -^r-j  ~^  tt  are  proportional  to  X,  Y,  Z,  and  if  we  set 

'  3s    3s    9s  ^     ^  7      7      7 

Ts=P^^  Js=P^^  ds^P^^ 

where  -  denotes  the  radius  of  curvature  of  the  line  s,  then 
P 

3£         d-n         dl 

^  ds  OS  OS 

By  differentiating  (7)  with  respect  to  s,  we  obtain 

dX         dY        dZ 

We  can  easily  show  that  -^7  - — 7  -r —  also  are  proportional  to  X,  Y,  Z.     In  fact, 
[from  10]   the  values  of  these  quantities  are  also   [equal  to] 

dZ        dY  dX     ^dZ         ^dY      dX 

'^'d7~^'di'       ^Tf~^^'       ^'di     "^  ds' 


therefore 


=  0, 

and  likewise  the  others.      We  set,  therefore, 


whence 


f=yx.      ^iL^.r,      >4^,z, 


,^.^(^)\(^)\(f)' 


NEW  GENERAL  INTESTIGATIONS  OF  CURVED  SURFACES  [1825]         109 

and  also 

5f'  dW  dU 

P'  =X-^  +  Y^  +Z^- 
^  ds  ds  ds 

Further   [we  obtain],  from  the  result  obtained  by  diflferentiating  (8), 
But  we  can  derive  two  other  expressions  for  this.      We  have 


[d  mri  __dri 


ds      ar      L  as       ar         ds       arJ 

therefore  [because  of  (8)] 

af  dr)  dt, 

^P'=^dd^^dd^^dd' 

[and  therefore,  from  (7),] 

,        dX         dY         dZ 

After  these  preliminaries  [using  (2)  and  (4)]  we  shall  now  first  put  m  in  the  form 

"^-^  dB^^  dQ^^  dff 
and  differentiating  with  respect  to  s,  we  have* 

dm  _  dx    d^'       dy    dr)'       ds     dCf 

"a7~a^'~a7a^"a7a^'"a7 
d'^x  av  d^s 

^^  ds.dd^^  ds.dd^''  ds.dd 
=  mp'{^'X-V'q'Y+  i'Z) 

^^  dd^^  dd^^  dd 

~^  dd^^  de^^  dd' 


*  It  is  better  to  differentiate  ml      [In  fact  from  (2)  and  (4) 

therefore 

am  _dx     a^-^;     ,   dy     d^y    ,  ds      d'^s 


dO  ddds     dd  dOds     dd  ddds 

af  ^dy)  dCl 


.)ii 


110  KARL  FRIEDRICH  GAUSS 

If  we  differentiate  again  with  respect  to  s,  and  notice  that 

l±  =  ^il^.    etc 
and  that 

we  have 


W  +  ^'TF  +  ^'W/ 
.ax  .    .ar  ,  .  .dZ\ 


-a7=^^(^'W  +  ^'TF  +  ^'-w)+^'(^^r^  +  ^a^  +  ^a-( 


/ax       ar       azw    ax      ,ar        a^-v 
/    ax  ^  ,ar  ^^,azw^ax^    ^y ^dZ\ 


p.M  I-  _/ar  a^     ar  a^\       /a^ax     dZdX\       /dXdr     axar\ 

[But  if  the  surface  element 

7n  ds  dO 

belonging  to  the  point  x,  y,  z  be  represented  upon  the  auxiliary  sphere  of  unit  radius 
by  means  of  parallel  normals,  then  there  corresponds  to  it  an  area  whose  magnitude  is 

J      i'^'Y  ^Z       dY  dZ\  idZdX       dZdX\  JoXdY      dXdY\\ 

t^^Vas  dd       dd  dsf'^^\ds  dd       dd  ds)^"^\ds  dd       dd  dsU 

Consequently,  the  measure  of  curvature  at  the  point  under  consideration  is  equal  to 

1  a'w?  1 
m  as^  J 


NOTES  111 


NOTES. 

The  parts  enclosed  in  brackets  are  additions  of  the  editor  of  the  German  edition 
or  of  the  translators. 

"  The  foregoing  fragment,  Neue  allgemeine  Untersuchungen  ilher  die  kriimmen  Fldchen, 
differs  from  the  Disquisitiones  not  only  in  the  more  limited  scope  of  the  matter,  but 
also  in  the  method  of  treatment  and  the  arrangement  of  the  theorems.  There  [paper 
of  1827]  Gauss  assumes  that  the  rectangular  coordinates  x,  y,  2  of  a  point  of  the  sur- 
face can  be  expressed  as  functions  of  any  two  independent  variables  p  and  q,  while 
here  [paper  of  1825]  he  chooses  as  new  variables  the  geodesic  coordinates  s  and  6. 
Here  [paper  of  1825]  he  begins  by  proving  the  theorem,  that  the  sum  of  the  three 
angles  of  a  triangle,  which  is  formed  by  shortest  lines  upon  an  arbitrary  curved  surface, 
differs  from  180°  by  the  area  of  the  triangle,  which  corresponds  to  it  in  the  represen- 
tation by  means  of  parallel  normals  upon  the  auxiliary  sphere  of  unit  radius.  From 
this,  by  means  of  simple  geometrical  considerations,  he  deriA^es  the  fundamental  theo- 
rem, that  "  in  the  transformation  of  surfaces  by  bending,  the  measure  of  curvature  at 
every  point  remains  unchanged."  But  there  [paper  of  1827]  he  first  shows,  in  Art. 
11,  that  the  measure  of  curvature  can  be  expressed  simply  by  means  of  the  three 
quantities  E,  F,  G,  and  their  derivatives  with  respect  to  p  and  q,  from  which  follows 
the  theorem  concerning  the  invariant  property  of  the  measure  of  curvature  as  a  corol- 
lary ;  and  only  much  later,  in  Art.  20,  quite  independently  of  this,  does  he  prove  the 
theorem  concerning  the  sum  of  the  angles  of  a  geodesic  triangle." 

Remark  by  Stackel,  Gauss's  Works,  vol.  viii,  p.  443. 

Art.  3,  p.  84,  1.  9.  cos^c^,  etc.,  is  used  here  where  the  German  text  has  coscfr, 
etc. 

Art.  3,  p.  84,  1.  13.    p'^,  etc.,  is  used  here  where  the  German  text  has  ^js,  etc. 

Art.  7,  p.  89,  11.  13,  21.  Since  XL  is  less  than  90°,  cosXL  is  always  positive 
and,  therefore,  the  algebraic  sign  of  the  expression  for  the  volume  of  this  pyramid 
depends  upon  that  of  sin  L'L".  Hence  it  is  positive,  zero,  or  negative  according  as 
the  arc  L'L"  is  less  than,  equal  to,  or  greater  than  180°. 

Art.  7,  p.  89,  11.  14-21.     As  is  seen  from  the  paper  of  1827  (see  page  6),  Gauss 


112  NOTES 

corrected  this  statement.  To  be  correct  it  should  read  :  for  which  we  can  write  also, 
according  to  well  known  principles  of  spherical  trigonometry, 

sin.  LL' .  smL' .  sin  L' L"=  sin  L' L"  .  sin  L"  .  sin  L" L^=  ^iuL" L  .  sin  X  .  sinZi', 

if  L,  L',  L"  denote  the  three  angles  of  the  spherical  triangle,  where  L  is  the  angle 
measured  from  the  arc  LL"  to  LL' ,  and  so  for  the  other  angles.  At  the  same  time 
we  easUy  see  that  this  value  is  one-sixth  of  the  pyramid  whose  angular  points  are 
the  centre  of  the  sphere  and  the  three  points  L,  L',  L" ;  and  this  pyramid  is  positive 
when  the  points  L,  L',  L"  are  arranged  in  the  same  order  about  this  triangle  as  the 
points  (1),  (2),  (3)  about  the  triangle  (1)  (2)  (3). 

Art.  8,  p.  90,  1.  7  fr.  bot.  In  the  German  text  V  stands  for  /  in  this  equation 
and  in  the  next  line  but  one. 

Art.  11,  p.  93,  1.  8  fr.  bot.  In  the  German  text,  in  the  expression  for  B,  (a/3'  +  a^') 
stands  for  (a'/8+  a/8'). 

Art.  11,  p.  94,  1.  17.  The  vertices  of  the  triangle  are  M,  M,  (3),  whose  coor- 
dinates are  a,  j3,  y;  a',  y8',  y' ;  0,  0,  1,  respectively.  The  pole  of  the  arc  MM'  on 
the  same  side  as  (3)  is  L,  whose  coordinates  are  X,  Y,  Z.  Now  applying  the  formula 
on  page  89,  line  10, 

x'y"-y'x"  =  sinZ'i"  cos  X(3), 
to  this  triangle,  we  obtain 

a ^'  -  /8  a'  =  sin  MM  cos  ^(3) 
or,  since 

itfilf  =  90°,  and  cosZ(3)=±Z 
we  have 

a;8'-/3a'  =  ±Z. 

Art.  14,  p.  100,  1.  19.  Here  X,  Y,  Z;  i,  r),  C,  0,  0,  1  take  the  place  of  a;,f/,s; 
ocf ,  y',  z' ;  x",  y",  s"  of  the  top  of  page  89.  Also  (3),  X  take  the  place  of  L',  L",  and 
^  is  the  angle  L  in  the  note  at  the  top  of  this  page. 

Art.  14,  p.  101,  1.  2  fr.  bot.  In  the  German  text  {^X- -q  XYZ  +  ^  Y^ Z\  stands 
for  \iX+-qXYZ-^Y'Z\. 

Art.  15,  p.  102,  1.  13  and  the  following.  Transforming  to  polar  coordinates, 
r,  6,  \fi,  by  the  substitutions  (since  on  the  auxiliary  sphere  r  —  l) 

X=  sin  ^sin»/;,        Y=  s'md  cosxp,       Z=gos0, 
c?JC=sin  6  cos  »/»  rf»/«+  cos  6  sini/;rf^,  dY—  —  sin  9  sin  \pd\p  +  cos  6  cos  xpdd, 

(1)  —  1 y2  (XdY—YdX)  becomes  cos  6  dxp. 


NOTES  113 

In  the  figures  on  page  102,  PL  and  P'L'  are  arcs  of  great  circles  intersecting  in 
the  point  (3),  and  the  element  LL' ,  which  is  not  necessarily  the  arc  of  a  great  circle, 
corresponds  to  the  element  of  the  geodesic  line  on  the  curved  surface.  (2)PP'(1) 
also  is  the  arc  of  a  great  circle.  Here  P'P  =  d\^,  .^=  cos  ^=  Altitude  of  the  zone 
of  which  LL'  P'P  is  a  part.  The  area  of  a  zone  varies  as  the  altitude  of  the  zone. 
Therefore,  in  the  case  under  consideration, 

Area  of  zone  _  Z 
2^r  ~T* 

Also 

KXQ&LL'P'P  _  drp 
Area  of  zone         2tt 
From  these  two  equations, 

(2)  Area  L  L'P'P  =  Zdyjj,  or  cos  6  d^. 

From  (1)  and  (2) 

-  j^2  {Xd  r-  YdX)  =  Area  L L'P'P. 

Art.  15,  p.  102.  The  point  (3)  in  the  figures  on  this  page  was  added  by  the 
translators. 

Art.  15,  p.  103,  U.  6-9.  It  has  been  shown  that  d(f)  =  Avea,LL'P'P,  =  dA,  say. 
Then 

<j>'  A 

J  d  (f>  =J  d  A, 
<f>  0 

or 

<f>'  —  cl)=A,  the  finite  area  L L'P'P. 

Art.  15,  p.  103,  1.  10  and  the  following.  Let  A,  B',  B^  be  the  vertices  of  a 
geodesic  triangle  on  the  curved  surface,  and  let  the  corresponding  triangle  on  the 
auxiliary  sphere  be  LL' L'^L,  whose  sides  are  not  necessarily  arcs  of  great  circles.  Let 
A,  B',  B^  denote  also  the  angles  of  the  geodesic  triangle.  Here  B'  is  the  supple- 
ment of  the  angle  denoted  by  B  on  page  103.  Let  <^  be  the  angle  on  the  sphere 
between  the  great  circle  arcs  L\,L[?>),i.e.,  ^  =  (3) XX,  X  coi'responding  to  the  dii-ec- 
tion  of  the  element  at  A  on  the  geodesic  line  AB',  and  let  ^' =  (3)Z'Xp  Xj  correspond- 
ing to  the  direction  of  the  element  at  B'  on  the  line  AB'.      Similarly,  let  »/(=  {S)Lfi, 


114 


NOTES 


i/»'  =  (3)  L\  jMp  [I,,  /Aj  denoting  the  directions  of  the  elements  at 
A,  B^,  respectively,  on  the  line  AB^.  And  let  x^^i^)L' v, 
;^j=  (3)i'iZ/j,  V,  v^  denoting  the  directions  of  the  elements  at 
B' ,  B^,  respectively,  on  the  line  B'  By 

Then  from  the  first  formula  on  page  103, 

<^'-<^  =  AreaZi:'P'P, 
i'  —  t|»  =  Area  LL\P\P, 
X,-x  =  Areai;'Z\P\P', 


JS! 


F'P/ 


i//'  -  i//  -  (<^'  -  (^)  -  (Xi -  x)  =  Area  LL\P\P-  Area  L L'  P'P -  Area  L'L\ P\P', 


or 

(1) 


(<^-^)  +  (x-f )  +  (f  -xJ  =  AreaiX'jZ'X. 

Since  X,  /a  represent  the   directions  of  the  linear  elements  at  A  on   the  geodesic 
lines  AB',  AB^  respectively,  the  absolute  value  of  the  angle  A  on  the   surface  is  meas- 
ured  by  the  arc  fi\,  or   by  the   spherical   angle  [xLX.      But  </>  —  »//  =  (3)ZX  —  (3)X/a 
=  jjiLX. 
Therefore 

Similarly 


180° 


A  =  <j)~xlj. 
-B'  =  -{x- 

A=f-Xr 


n 


Therefore,  from  (1), 

A +B' +  B^-180°  =  Area  LL\L'L. 

Art.  15,  p.  103,  1.  19.  In  the  German  text  i^X'P'P  stands  iox  LL\P\P, 
which  represents  the  angle  v/*'  — 1/». 

Art.  15,  p.  104,  1.  12.      This  general  theorem  may  be  stated  as  follows : 

The  sum  of  all  the  angles  of  a  polygon  of  n  sides,  which  are  shortest  lines 
upon  the  curved  surface,  is  equal  to  the  sum  of  (?z  — 2)180°  and  the  area  of  the 
polygon  upon  the  auxiliary  sphere  whose  boundary  is  formed  by  the  points  L  which 
correspond  to  the  points  of  the  boundary  of  the  given  polygon,  and  in  such  a  manner 
that  the  area  of  this  polygon  may  be  regarded  positive  or  negative  according  as  it  is 
enclosed  by  its  boundary  in  the  same  sense  as  the  given  figure  or  the  contrary. 

Art.  16,  p.  104,  1.  12  fr.  bot.  The  ze^iith  of  a  point  on  the  surface  is  the  cor- 
responding point  on  the  auxiliary  sphere.  It  is  the  spherical  representation  of  the 
point. 

Art.  18,  p.  110,  1.  10.  The  normal  to  the  surface  is  here  taken  in  the  direction 
opposite  to  that  given  by   [9]  page  107. 


BIBLIOGRAPHY 


BIBLIOGRAPHY. 

This  bibliography  is  limited  to  books,  memoirs,  etc.,  which  use  Gauss's  method  and  -which  treat,  more  or  less 
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surfaces,  deformation  of  surfaces,  orthogonal  systems,  and  the  general  theory  of  surfaces.  Several  papers  which  lie 
beyond  these  limitations  have  been  added  because  of  their  importance  or  historic  interest.  For  want  of  space,  gener- 
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BIELIOGRAPHT 


125 


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Memoire    sur    I'emploi   des    coordonnees   curviligneg 

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des  surfaces  isothermes.     Ann.  de  la  Soc.  soi.  de  Brux- 
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Memoire  sur  la  recherche  la  plus  generale  d'un  sys- 

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Theorie   nouvelle  du  systeme   orthogonal  triplement 

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satzes   fiir    Dreiecke    in  stetig    gekriimmten    Flachen. 

Nachr.  der  Kgl.   Gesell.   der  Wissenschaften  zu   Got- 

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Bemerkungen  zur  Gesohichte  der  geodatischen  Linien. 

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Beitrage  zur  Fliichentheorie.    Berichte  der  Kgl.  Gesell. 

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126 


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CORRIGENDA   BT   ADDENDA. 

Art.  11,  p.  20,  1.  6.      The  fourth  E  should  be  F. 

Art.  18,  p.  27,  1.  7.  For  V  {EG-F^)  .  dp  .  dO  read  2  i/{FG-F^)  .  dq  .  dd. 
The  original  and  the  Latin  reprints  lack  the  factor  2  ;  the  correction  is  made  in  all 
the  translations. 

Art.  19,  p.  28,  1.  10.      For  (/  read  q. 

Art.  22,  p.  34,  1.  5,  left  side;  Art.  24,  p.  36,  1.  5,  third  equation;  Art.  24, 
p.  38,  1.  4.      The  original  and  Liouville's  reprint  have  q  for  p. 

Note  on  Art.  23,  p.  55,  1.  2  fr.  hot.      For  p  read  q. 


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